pantheon.ufrj.br · A STUDY ON GREEN WATER PROBLEM WITH DAM BREAK MODEL AND THE BIV TECHNIQUE Qingfeng Duan TESE SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE POS-GRADUAC˘

A STUDY ON GREEN WATER PROBLEM WITH DAM BREAK MODEL AND

THE BIV TECHNIQUE

Qingfeng Duan

Tese de Doutorado apresentada ao Programa

de Pos-graduacao em Engenharia Oceanica,

COPPE, da Universidade Federal do Rio de

Janeiro, como parte dos requisitos necessarios

a obtencao do tıtulo de Doutor em Engenharia

Oceanica.

Orientador: Sergio Hamilton Sphaier

Rio de Janeiro

Abril de 2017


THE BIV TECHNIQUE

Qingfeng Duan

TESE SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ

COIMBRA DE POS-GRADUACAO E PESQUISA DE ENGENHARIA (COPPE)

DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS

REQUISITOS NECESSARIOS PARA A OBTENCAO DO GRAU DE DOUTOR

EM CIENCIAS EM ENGENHARIA OCEANICA.

Examinada por:

Prof. Sergio Hamilton Sphaier, Dr.-Ing.

Dr. Marcelo de Araujo Vitola, D.Sc.

Prof. Paulo de Tarso Themıstocles Esperanca, D.Sc.

Prof. Claudio Alexis Rodriguez Castillo, D.Sc.

Prof. Su Jian, D.Sc.

Prof. Luis Fernando Alzuguir Azevedo, Ph.D.

RIO DE JANEIRO, RJ – BRASIL

ABRIL DE 2017

Duan, Qingfeng

A study on green water problem with dam break model

and the BIV technique/Qingfeng Duan. – Rio de Janeiro:

UFRJ/COPPE, 2017.

XVII, 136 p.: il.; 29, 7cm.


Tese (doutorado) – UFRJ/COPPE/Programa de

Engenharia Oceanica, 2017.

Bibliografia: p. 111 – 119.

1. Green water. 2. Dam break. 3. BIV. I.

Sphaier, Sergio Hamilton. II. Universidade Federal do Rio

de Janeiro, COPPE, Programa de Engenharia Oceanica.

III. Tıtulo.

iii

To my dear family, who always

picked me up and encourage me

to go on every adventure,

especially this one.

iv

Acknowledgement

Firstly, I would like to express my sincere gratitude to my advisor Prof. Sergio

Hamilton Sphaier for the continuous support of my Ph.D. study and research, for

his patience, motivation, and enthusiasm.

I would also like to express my special appreciation and thanks to Prof. Marcelo

de Araujo Vitola for his precious critical comments in many aspects of my research.

The door to his office was always open whenever I ran into a trouble spot or had a

question about my research.

Many thanks to Prof. Paulo de Tarso T. Esperanca and to the fellow mates in

LabOceano: Jassiel Hernandez, Fabio Quintana, Alex Alves and all others. Very

grateful for all the help in the experiment.

Special thanks to all my friends, especially for Dr. Guang Ming Fu and Prof. Su

Jian. Thank you for listening, offering me advice, and supporting me through this

entire process. It is all of you colored my life in Rio de Janeiro.

My thanks also go to the administrative staff in PENO for all the help during

my Ph.D.

I also take this opportunity to record my sincere gratitude to one and all who,

directly or indirectly, have lent their helping hands in this adventure.

Besides, financial supports from the China Scholarship Council (CSC), the

Brazilian National Council for the Improvement of Higher Education (CAPES) and

the Brazilian National Council for Scientific and Technological Development (CNPq)

are gratefully acknowledged.

v

Resumo da Tese apresentada a COPPE/UFRJ como parte dos requisitos necessarios

para a obtencao do grau de Doutor em Ciencias (D.Sc.)

UM ESTUDO SOBRE O PROBLEMA DA AGUA VERDE COM MODELO DE

QUEBRA DE BARRAGEM E A TECNICA BIV

Qingfeng Duan

Abril/2017


Programa: Engenharia Oceanica

A agua verde gerada por ondas extremas e principalmente uma mistura ar-agua,

que inclui uma fase contınua e uma fase dispersa.

A fase contınua e estudado principalmente com o modelo de quebra de barragem

(QB) por tres abordagens principais, a saber analıtica, experimental e numerica.

Analiticamente, as duas solucoes de Stoker sao introduzidas para estudar a onda de

QB em um canal horizontal seco. Nas fases iniciais, o conceito de “ruptura repentina

de barragens” e revisto. A barragem pode ser considerada quebra repentinamente

quando o perodo de remocao adimensional e menor que3t∗15≈ 0.63. Para incluırem

os efeitos de atrito e de inclinacao de fundo, propusemos uma solucao fragmentada

simplificada (PS). Numericamente, o modelo de dois fluidos com o metodo Volume

of Fluid (VOF) implementado no solver interFoam e usado para simular a onda de

QB. O solucionador interFoam resolve as equacoes de RANS com um modelo de tur-

bulencia k− ε. Experimentalmente, diferentes casos de sao realizadas incluindo um

leito horizontal e quatro leito como diferentes angulo de inclinacao (ascendente 10◦ e

5◦, descendente 10◦ e 5◦). Ao introduzir um fator de compensacao de tempo t∗c =t∗13

em PS, a onda de downstream prevista concorda bem com os dados experimentais

e os resultados numericos.

Na fase dispersa, os casos presentes aplicaram principalmente a tecnica BIV. Para

melhorar a compreensao da tecnica BIV, sao consideradas as propriedades especiais

das bolhas e e realizado um experimento BIV de teste para avaliar a capacidade

do sistema BIV. Finalmente, a tecnica e aplicada no experimento de ruptura da

barragem.

vi

Abstract of Thesis presented to COPPE/UFRJ as a partial fulfillment of the

requirements for the degree of Doctor of Science (D.Sc.)


THE BIV TECHNIQUE

Qingfeng Duan

April/2017

Advisor: Sergio Hamilton Sphaier

Department: Ocean Engineering

The green water generated by extreme waves is primarily an air-water mixture,

which includes a continuous phase and a dispersed phase.

The continuous phase is explored with the dam break model by three main ap-

proaches, namely analytical, experimental and numerical. Analytically, both the

Eulerian and Lagrangian Stoker solutions are introduced to study the dam break

wave in a dry horizontal channel. The two Stoker solutions have an intersection

point at the dam section at instant t∗1 =√

103

. In the initial stages, the “sudden

dam break” concept is revised. The dam may be considered break suddenly when

the dimensionless gate removal period is smaller than3t∗15≈ 0.63. To include the

bottom friction effects and the bed slope effects for green water problem, we pro-

posed a simplified piecewise solution (PS). Numerically, the two fluid model with

the Volume of Fluid (VOF) method implemented in the interFoam solver is used

to simulate the dam break wave. The interFoam solves the RANS equations with

a k − ε turbulence model. Experimentally, different dam break cases are carried

out including one horizontal bed and four different downstream slope beds (upward

10◦ and 5◦, downward 10◦ and 5◦). Each bed is tested with two different water

levels (110 mm and 220 mm). By introducing a time compensation factor t∗c =t∗13

into PS, the predicted downstream wave agree well with the experimental data and

numerical results.

In the dispersed phase, the present study mainly focus on the BIV technique. The

special properties of bubbles are taken into consideration and a test BIV experiment

is carried out to evaluate the BIV system capacity. Finally, the BIV technique is

applied to the dam break experiment.

vii

Contents

List of Figures xi

List of Tables xv

1 Introduction 1

1.1 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature review 7

2.1 The green water phenomenon . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Scenarios of green water . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Green water impact on structures . . . . . . . . . . . . . . . . 9

2.1.3 Green water loading on deck . . . . . . . . . . . . . . . . . . . 11

2.2 The BIV technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 General Description of PIV . . . . . . . . . . . . . . . . . . . 12

2.2.2 Consideration for PIV in bubbly flow . . . . . . . . . . . . . . 15

2.2.3 General description of BIV technique . . . . . . . . . . . . . . 17

3 Analytical Dam Break Solutions 19

3.1 Stoker solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Eulerian Stoker solution . . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Lagrangian Stoker solution . . . . . . . . . . . . . . . . . . . . 20

3.2 Eulerian Stoker solution extensions . . . . . . . . . . . . . . . . . . . 24

3.2.1 Bottom friction effects . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Dam break wave in a sloping channel . . . . . . . . . . . . . . 26

3.3 A piecewise solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Solution with a compensated time factor . . . . . . . . . . . . 29

4 Numerical Dam Break Model 31

4.1 Overview of OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 interFoam solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 RANS equations . . . . . . . . . . . . . . . . . . . . . . . . . 32

viii

4.2.2 Modified VOF method . . . . . . . . . . . . . . . . . . . . . . 35

4.2.3 Discretised model equations . . . . . . . . . . . . . . . . . . . 36

4.2.4 Pressure-velocity coupling . . . . . . . . . . . . . . . . . . . . 38

4.3 Validation of the solver . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 Grid convergence . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Dam Break Experiments 45

5.1 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Tank and gate release system . . . . . . . . . . . . . . . . . . . . . . 46

5.3 Data acquisition and synchronization . . . . . . . . . . . . . . . . . . 47

5.3.1 Wave probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3.2 Pressure sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3.3 Temperature sensors . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.4 Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.5 Video recording and illumination . . . . . . . . . . . . . . . . 51

5.4 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Results of Dam Break Models 54

6.1 Main period with Stoker solutions . . . . . . . . . . . . . . . . . . . . 54

6.1.1 Sudden dam break . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Main period with the piecewise solution . . . . . . . . . . . . . . . . . 61

6.2.1 Horizontal bed cases . . . . . . . . . . . . . . . . . . . . . . . 61

6.2.2 Slope bed cases . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2.3 Bottom friction factor . . . . . . . . . . . . . . . . . . . . . . 73

6.3 Pressure measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3.1 Typical signal of pressure . . . . . . . . . . . . . . . . . . . . 76

6.3.2 Numerical pressure results . . . . . . . . . . . . . . . . . . . . 81

6.4 Elevations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.1 Wave probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.2 Virtual wave probes . . . . . . . . . . . . . . . . . . . . . . . 85

7 The BIV technique and the Application 87

7.1 BIV test experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.1.1 System capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.2 Considerations on bubble/drops properties . . . . . . . . . . . . . . . 93

7.2.1 Shape regimes of the fluid particles . . . . . . . . . . . . . . . 94

7.2.2 Equation of particle motion . . . . . . . . . . . . . . . . . . . 97

7.3 Application of the BIV technique . . . . . . . . . . . . . . . . . . . . 101

7.3.1 Example in the main period . . . . . . . . . . . . . . . . . . . 102

7.3.2 Example in the impact period . . . . . . . . . . . . . . . . . . 105

ix

7.3.3 Example in the reflection period . . . . . . . . . . . . . . . . . 106

7.3.4 Notes on BIV technique application . . . . . . . . . . . . . . . 106

8 Conclusions and Future Work 108

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.1.1 Continuous phase . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.1.2 Dispersed phase . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Bibliography 111

A Shallow Water Equations 120

A.1 Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . . . . . 120

A.1.1 Saint Venant Equations . . . . . . . . . . . . . . . . . . . . . 123

A.1.2 Method of Characteristics . . . . . . . . . . . . . . . . . . . . 123

B Camera Calibration 124

B.1 camera calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C Water Level Dectection 127

C.1 Algorithm for water level detection . . . . . . . . . . . . . . . . . . . 127

D Uncertainty Analysis of BIV 132

D.1 Uncertainty analysis of BIV . . . . . . . . . . . . . . . . . . . . . . . 132

D.1.1 Optimize magnification factor . . . . . . . . . . . . . . . . . . 135

x

List of Figures

1.1 Green water incident of the Selkirk Settler. Pho-

tograph by Captain G. A. Ianiev. Available at:

http://www.boatnerd.com/pictures/fleet/spruceglen.htm [accessed

17th April 2017] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Main methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Scenarios of green water. Adapted from GRECO et al. [24] . . . . . . 8

2.2 Schematic green water flow over the deck . . . . . . . . . . . . . . . . 9

2.3 Results of similarity solution by ZHANG et al. [27], adapted from

GRECO [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Typical arrangement for particle image velocimetry . . . . . . . . . . 12

2.5 The cross-correlation algorithms, adapted from RAFFEL et al. [16] . 14

2.6 Principle of the three-point Gaussian fit, adapted from THIELICKE

and STAMHUIS [38] . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 PIV image around an air bubble in a thin light-sheet, adapted from

BRUCKER [39] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.8 Reflection and refraction of components of an incident ray for an air

bubble, adapted from DAVIS [40] . . . . . . . . . . . . . . . . . . . . 16

2.9 Sketch of a combination of the shadowgraph with PIV, adapted from

SATHE et al. [47] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.10 Sketch of BIV arrangements, adapted from RYU et al. [15] . . . . . . 17

3.1 Dam break in Eulerian representation, adapted from STOKER [10] . 20

3.2 Dam break in Lagrangian representation, adapted from STOKER [10] 21

3.3 The Whitham solution . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Sketch of a dam break wave in a dry downward sloping channel,

adapted from CHANSON [53] . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Sketch of a dam break wave in a dry upward sloping channel, adapted

from CHANSON [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Sketch of a horizontal dam with different sloping bed . . . . . . . . . 28

3.7 Example of transformation for θ = 10 and f = 0.1 at instant t∗ = 0.667 29

xi

4.1 Discretization of the solution domain, adapted from GREEN-

SHIELDS [63] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Schematic of dam break model,in units: mm, adapted from

LOBOVSKY et al. [32] . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Results of pressure comparing to experimental data . . . . . . . . . . 43

5.1 Experimental setup, overall view . . . . . . . . . . . . . . . . . . . . . 45

5.2 Experimental setup of the dam-break tests . . . . . . . . . . . . . . . 46

5.3 Schematic view of the gate release system . . . . . . . . . . . . . . . 47

5.4 Data acquisition Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 Downsteam turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 Wave probes and virtual wave probes . . . . . . . . . . . . . . . . . . 49

5.7 Sketch of the slope bed blocks . . . . . . . . . . . . . . . . . . . . . . 53

6.1 H=110 mm, evaluation of free surface profile . . . . . . . . . . . . . . 55

6.2 OpenFOAM free surface profile . . . . . . . . . . . . . . . . . . . . . 56

6.3 Sketch of the initial stage of the dam break problem with top particle

P1 and bottom particle P2 at the dam gate section . . . . . . . . . . . 57

6.4 H=300 mm, evaluation of free surface profile, adapted from from

LOBOVSKY et al. [32] . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.5 Results of the two higher level cases at instant tg . . . . . . . . . . . 60

6.6 Example results of the piecewise solution in horizontal channel . . . . 61

6.7 H=110 mm, results of the PS with compensated time t∗c =t∗13

, and

with bottom friction factor f = 0.04 . . . . . . . . . . . . . . . . . . 62

6.8 H=220 mm, results of the PS with compensated time t∗c =t∗13

, and

with bottom friction factor f = 0.12 . . . . . . . . . . . . . . . . . . 63

6.9 H=600 mm, adapted from LOBOVSKY et al. [32], results of the PS

with compensated time t∗c =t∗13

, and with bottom friction factor f =

0.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.10 H=110 mm, evaluation of free surface profile for downward slope

angle 10 degree, with t∗c =t∗13

and with f = 0.04 . . . . . . . . . . . . 66



and with f = 0.12 . . . . . . . . . . . . 67



and with f = 0.04 . . . . . . . . . . . . . 68



and with f = 0.12 . . . . . . . . . . . . . 69



and with f = 0.04 . . . . . . . . . . . . 70

xii

6.15 H=220 mm, evaluation of free surface profile for upward slope angle

10 degree, with t∗c =t∗13

and with f = 0.12 . . . . . . . . . . . . . . . 71



and with f = 0.04 . . . . . . . . . . . . 72



and with f = 0.12 . . . . . . . . . . . . 73

6.18 A typical signal of pressure sensors and the zoomed pressure peaks . . 76

6.19 Small peaks before impact related to the gate releasing . . . . . . . . 77

6.20 Pitch motion test with 6 DOF platform . . . . . . . . . . . . . . . . . 77

6.21 Pitch motion test in different periods, the amplitude of the pitch angle

is 10 degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.22 Pressure peaks in ten times repetitions for horizontal bed case H =

110 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.23 Simulated pressure for H = 220 mm . . . . . . . . . . . . . . . . . . . 81

6.24 Simulated pressure for H = 110 mm . . . . . . . . . . . . . . . . . . . 82

6.25 Elevation time history of wave probe h7 and h8 for horizontal bed

case H = 110 mm, in 10 repetitions . . . . . . . . . . . . . . . . . . . 82

6.26 Elevation time history at wave probe location h7 and h8 for horizontal

bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.27 Elevation time history at wave probe location h7 and h8 . . . . . . . 84

6.28 Elevation of virtual wave probe h5 . . . . . . . . . . . . . . . . . . . 85

6.29 Water Level Dectection of h5 (the blue one) at time 120 ms (left) and

132 ms (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.30 Elevation of virtual wave probe h3 (left) and the corresponded areated

peak at time 224 ms (right) . . . . . . . . . . . . . . . . . . . . . . . 86

6.31 Elevation of virtual wave probe h1 (left) and the corresponded small

areated front at time 212 ms (right) . . . . . . . . . . . . . . . . . . . 86

7.1 Schematic of the BIV test experiment . . . . . . . . . . . . . . . . . . 87

7.2 An example image and bubble identify process . . . . . . . . . . . . . 89

7.3 Workflow of PIVlab, adapted from THIELICKE and STAMHUIS [38] 90

7.4 An example of the output velocity fields . . . . . . . . . . . . . . . . 91

7.5 Results for the test experiment . . . . . . . . . . . . . . . . . . . . . 93

7.6 Shape regimes for bubbles and drops, adapted from CLIFT et al. [81] 94

7.7 Bond number versus bubbles or drops diameter . . . . . . . . . . . . 96

7.8 Terminal Reynolds number versus bubbles or drops diameter . . . . . 97

7.9 Velocity amplitude response versus Stokes number . . . . . . . . . . . 100

7.10 H=110 mm, BIV analysis velocity vectors in time 12, 20, 28, 36, 44,

52 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xiii

7.11 Gate releasing velocity in time history obtained with BIV . . . . . . . 103

7.12 BIV results example during the runup . . . . . . . . . . . . . . . . . 103

7.13 Wave front velocity time evolution . . . . . . . . . . . . . . . . . . . . 104

7.14 BIV results example during the runup . . . . . . . . . . . . . . . . . 106

7.15 BIV results example of the reflection wave . . . . . . . . . . . . . . . 106

A.1 A Typical Water Column . . . . . . . . . . . . . . . . . . . . . . . . . 121

B.1 camera calibration image example . . . . . . . . . . . . . . . . . . . . 124

B.2 camera calibration mean error . . . . . . . . . . . . . . . . . . . . . . 125

B.3 camera calibration mean error . . . . . . . . . . . . . . . . . . . . . . 125

C.1 Original . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

C.2 Edge dectected with Sobel method . . . . . . . . . . . . . . . . . . . 128

C.3 Edge thickness dilated . . . . . . . . . . . . . . . . . . . . . . . . . . 129

C.4 Hough transform for straight lines . . . . . . . . . . . . . . . . . . . . 129

C.5 Hough transform peaks . . . . . . . . . . . . . . . . . . . . . . . . . . 130

C.6 Dectected top three longest line . . . . . . . . . . . . . . . . . . . . . 130

C.7 Calculate the detected water level . . . . . . . . . . . . . . . . . . . . 131

xiv

List of Tables

3.1 Summary of the piecewise solution . . . . . . . . . . . . . . . . . . . 29

4.1 The discretization schemes of interFoam . . . . . . . . . . . . . . . . 42

4.2 Order of accuracy and Grid Convergence Index for three integration

variables. Subscripts 3, 2 and 1 represent case DS = 20, 10, 5 mm,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1 Specifications of EPX sensor . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Camera properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Slope bed blocks and test cases . . . . . . . . . . . . . . . . . . . . . 53

6.1 The average gate releasing velocity and the gate removal period . . . 60

6.2 Summary of the different bed slope bed cases . . . . . . . . . . . . . 65

6.3 Bottom friction factor calculation with Approach A . . . . . . . . . . 74

6.4 Bottom friction factor calculation with Approach B . . . . . . . . . . 75

7.1 Summary of BIV test experiment . . . . . . . . . . . . . . . . . . . . 90

7.2 Analytical approaches of front velocity with bottom friction effects . . 104

7.3 The average dimensionless wave front velocity for time t > t∗1 . . . . . 105

D.1 Evaluate the uncertainty on α . . . . . . . . . . . . . . . . . . . . . . 133

D.2 Evaluate the uncertainty on image displacement ∆X . . . . . . . . . 134

D.3 Evaluate the uncertainty on δu . . . . . . . . . . . . . . . . . . . . . 134

D.4 Evaluate the uncertainty on ∆t . . . . . . . . . . . . . . . . . . . . . 135

D.5 Evaluate the uncertainty on u . . . . . . . . . . . . . . . . . . . . . . 135

xv

Nomenclature

Abbreviations

2D Two Dimensional

3D Three Dimensional

BIV Bubble Image Velocimetry

DCC Direct Cross Correlation

DFT Discrete Fourier Transform

DOF Depth of Field

DSR Dynamic Spatial Range

DVR Dynamic Velocity Range

ESS Eulerian Stoker Solution

FFT Fast Fourier Transform

FOV Field of View

LHS Left Hand Side

LSS Lagrangian Stoker Solution

OpenFOAM Open Field Operation And Manipulation

PIV Particle Image Velocimetry

RANS Reynolds Averaged NavierStokes

RHS Right Hand Side

VOF Volume of Fluid

Symbols

xvi

α Water volume fraction

κ Kinematic viscosity

µ Dynamic viscosity

µeff Effective dynamic viscosity

ν Kinematic viscosity

ρ Density

θ Angle

Bo Bond number

dp Particle diameter

f Bottom friction factor

g Gravitational constant

H Initial water level in dam

h Local water depth

Mo Morton number

Re Reynolds number

ReT Reynolds number in terminal velocity

St Stokes number

t Time

t1 Time for ESS and LSS has an intersection point at dam section

tc Time compensation factor

tg Time at gate releasing distance equals to H

U Dam break wave front Velocity

u Horizontal velocity component

UT Terminal velocity

v Horizontal velocity component

x Horizontal axis

y Vertical axis

xvii

Chapter 1

Introduction

Interactions between extreme waves and offshore structures are of primary interest

to ocean engineers. In rough sea conditions, when the relative motion between the

ship deck and the local water surface becomes so excessive that the incoming wave

can overcome the freeboard and flow onto the deck of a ship. This problem is known

as “shipping of water”, “deck wetness” or “green water”.

(a) Before incident (b) During incident

Figure 1.1: Green water incident of the Selkirk Settler. Photograph by Captain G.A. Ianiev. Available at: http://www.boatnerd.com/pictures/fleet/spruceglen.htm[accessed 17th April 2017]

Green water has been considered to be a major issue with regards to safety

and operability of offshore structures for a long time. SCHØNBERG and RAINEY

[1] reported a green water accident for the Selkirk Settler in the mid Atlantic in

February 1987, as shown in Fig. 1.1, the upper deck is being swamped by a huge

wave crest. That same storm sank two ships and severely damaged three others.

In the oil and gas industry, as the demand for fossil fuel is increasing, oil produc-

tion has moved into the deep water. A common deep water facility is the floating

production storage and offloading (FPSO) vessel. FPSO units are the ship-type

structure, often permanently moored to the seabed. They remain at their position

1

during a storm and their decks carry a lot of sensitive equipment, which is at the

risk of possible damage due to the shipping of water on deck. One example of a

critical green water event of the Norwegian FPSO, Petrojarl Varg, was reported by

ERSDAL et al. [2] in January 2000. The living quarters were hit by green water and

resulted in the damage of a window at the second floor. LEONHARDSEN et al. [3]

reported two green water incidents in the North Sea which forced the shutdown of

production and damaged safety equipment. MORRIS et al. [4] reported that, from

year 1995 to year 2000, 17 green water incidents were identified on 12 UK FPSO

units and caused different level of damage. Due to its importance in oil production,

the interests on the green water problem are growing.

Green water generated by extreme waves is primarily an air-water mixture, as

can be observed in Fig. 1.1b. It includes a continuous phase and a dispersed phase.

In the continuous phase, since the resemblance between the dam break model and

the green water phenomenon was reported by GODA et al. [5], the dam break model

has become the most popular approach for representing the shipping of water on

deck, e.g. BUCHNER [6], GRECO et al. [7], SCHØNBERG and RAINEY [1], RYU

et al. [8], PHAM [9]. The standard approach for solving the dam break problem is

the so-called “Stoker solution” (STOKER [10]).

The “Stoker solution” is a simplified solution, which restricts to a horizontal bed

case without bottom friction. However, during a green water incident, the pitch

angle of ship deck varies due to the incoming wave. According to BUCHNER [6]

and GRECO [11], the green water impact on structures are determined by the flow

velocity and its propagating wave front shape, where the bottom friction taking

place. The “Stoker solution” do not take this two aspects into consideration and

thus there is a calling on extending the solution to include the two aspects.

The initial stage of the dam break model is another interesting topic. Unlike its

application for hydraulics researchers usually with a long open channel, in shipping

of water, the deck of the ship is relatively short. The limited deck length makes the

initial stage of the dam break model more relevant. In the initial stage, there is a

discontinuity in the “Stoker solution”, the wave front velocity jumps from zero to

2√gH at t = 0+, where g is the gravity acceleration and H is the initial water level

within the dam. This indicates that the “Stoker solution” is not a good approxima-

tion for dam break wave in the initial stage. To the best of our knowledge, despite a

few studies, e.g. STANSBY et al. [12], BUKREEV and GUSEV [13] and OERTEL

and BUNG [14], the initial stage has not been well investigated yet.

On the other hand, the dispersed phase of the shipping of water has rarely

been studied. Due to the high void fraction nature of the green water flow, the

measurement on a important characteristics of flow, the velocity field, becomes

very difficult. Recently, RYU et al. [15] introduced the so-called Bubble Image

2

Velocimetry (BIV) technique, based on the idea of combining the shadowgraph

technique that illuminates the fluid from behind to reveal the flow pattern, and the

classical Particle Image Velocimetry (PIV) technique that correlates the consecutive

images to determine the velocity. In contrast to the PIV technique, no small seeding

particles are added in the flow, but the bubbles itself are used as the tracer particles.

Using the BIV technique, RYU et al. [8] succeed in measuring the velocity field

of several green water cases induced by breaking waves. They concluded that the

comparisons with the “Stoker solution” are surprisingly well. However, PIV is a very

dedicated technique which has many special requirements on the seeding particles.

Despite of the success achieved by RYU et al. [8] in obtaining the velocity field, a

very important aspect, the different properties between bubbles and solid seeding

particles, was not taken into consideration in their work. According to RAFFEL

et al. [16], the seeding particles should be small enough to follow the flow and also

be large enough to scatter sufficient light; the density of the particles should be close

to the fluid; the distribution of particles should be homogeneous. Comparing to the

solid seeding particles, the properties of bubbles are quite different. The density of

air bubbles is only one thousandth of water; the bubbles size and its distribution

depend on the flow itself, which are almost out of control; the bubbles may break-up

into small ones or coalescence into a bigger one. None of these specific properties of

bubbles has been considered in work by RYU et al. [8].

1.1 Research objectives

Based on the above observations, this thesis is aimed to improve the knowledge

of the green water problem on both phases. In the continuous phase, the main

objectives are:

• To extend the analytical dam break solution for green water problem by taking

account the bottom friction and the bed slope effects;

• To evaluate a numerical model based on open source code OpenFOAM;

• To do dam break experiments with different slope bed and acquire neces-

sary experimental data that can validate the analytical solution and numerical

model;

In the dispersed phase, the main objectives are:

• To test the BIV technique and evaluate its system capacity;

• To evaluate the special characteristics of bubbles that differ the PIV and BIV

technique;

3

• To acquire the dam break velocity fields in the dispersed phase using the BIV

technique;

In order to achieve these goals, the main methodology are sketched in Fig. 1.2a

and Fig. 1.2b respectively.

Continuous Phase

Dam Break Model

Analytical Numerical Experimental

ESS & LSS

Initial stages

Extended solution

interFOAM

VOF

RANS

One horizontal bed

Four slope beds

Two water levels

(a) Continuous phase, ESS donates Eulerian Stoker solution and LSS representsLagrangian Stoker solution

Dispersed Phase

BIV technique

Bubbles/drops properties System capacity Technique Application

Shape regimes

Velocity response

Uncertainty

DVR & DSR

Dam break experiment

(b) Dispersed phase, DSR donates dynamic spatial range and DVR representsdynamic velocity range

Figure 1.2: Main methodology

In the continuous phase, the dam break model is applied. The dam break model

is explored by three main approaches, namely analytical, experimental and numer-

ical. Two main topics of the dam break model are explored: the initial stage and

an extended solution to include bottom friction and bed slope effects. Analytically,

STOKER [10] proposed two dam break solutions using Eulerian and Lagrangian

representation. To distinguish the two solutions, they are called as the Eulerian

Stoker solution (ESS) and the Lagrangian Stoker solution (LSS) respectively. Both

Stoker solutions are introduced to studied the dam break wave in a dry horizon-

tal channel. In the initial stages, the “sudden dam break” concept proposed by

4

LAUBER and HAGER [17] is also investigated. To include the bottom friction

effects and the bed slope effects, a simplified piecewise solution (PS) is proposed

based on the work by WHITHAM [18] and CHANSON [19]. Numerically, the two

fluid model with the Volume of Fluid (VOF) method implemented in the interFoam

solver is used to simulate the dam break wave. The interFoam solver solves the

Reynolds-averaged Navier-Stokes equations (RANS) with a k− ε turbulence model.

Experimentally, different dam break cases are carried out including one horizontal

bed and four different downstream bed slopes (upward 10◦ and 5◦, downward 10◦

and 5◦). For each bed, the experiment is run with two different water levels (a lower

level 110 mm and a higher level 220 mm). The results are evaluated by comparing

different approaches.

In the dispersed phase, the present study is mainly focusing on the BIV tech-

nique, which requires a good knowledge on the special characteristics of bubbles.

With regarding to measurement accuracy, two main aspects of bubble characteristics

are considered: the bubble dimension size and bubble density. The bubble dimension

size are related with its shape regimes which are determined by three dimensionless

groups: the Bond number, the Morton number and the terminal Reynolds number.

The density ratio between the particles and the surrounding fluid affects the particle

velocity response in the fluid. This is investigated with the particle motion equa-

tion: the Basset-Boussinesq-Oseen (BBO) equation. To test the performance of the

BIV technique, a test experiment is carried out. A uncertainty analysis is done by

the ITTC Recommended Procedures and Guidelines on PIV uncertainty analysis

(PARK et al. [20]). The capacity of the BIV system, including the dynamic velocity

range (DVR) and dynamic spatial range (DSR) are also explored.

1.2 Outline of the thesis

The thesis is structured as follows.

After this brief introduction, the general physics of green water phenomenon

and the developments in the PIV technique and the BIV technique are reviewed in

Chapter 2.

In Chapter 3, the analytical approaches for the dam break model are explored.

The theoretical solutions proposed by STOKER [10] for a horizontal bed cases are

given first. To include the bottom friction effects and the bed slope effects, a sim-

plified piecewise solution (PS) is proposed based on the work by WHITHAM [18]

and CHANSON [19].

In Chapter 4, the two fluid model with the Volume of Fluid (VOF) method im-

plemented in the interFoam solver (version 2.3.0) is described in detail. The solver

solves the Reynolds-averaged Navier-Stokes equations (RANS) with a k − ε turbu-

5

lence model. The solver is also validated with previously published experimental

data for a horizontal bed case.

In Chapter 5, the dam break experiments are described. Different dam break

cases are carried out including one horizontal bed and four different downstream bed

slopes (upward 10◦ and 5◦, downward 10◦ and 5◦). For each bed, the experiment

is run with two different water levels (a lower level 110 mm and a higher level 220

mm).

In Chapter 6, the results are evaluated by comparing different approaches. For

the horizontal bed cases, the initial stages of the dam break problem are explored.

The “sudden dam break” concept proposed by Lauber and Hager is also investigated.

In Chapter 7, the BIV technique is explored. A test BIV experiment is carried

out first. The uncertainty and the capacity of the BIV system are explored. The

main difference with the PIV technique is that the BIV technique uses the bubbles

as tracer particles. The special characteristics on bubbles are also explored by the

bubble (or drops) shape regimes and velocity response in fluid. The BIV technique

is then applied to the dam break experiment.

Finally, the main conclusions are remarked in Chapter 8.

6

Chapter 2

Literature review

Green water has been considered to be a major issue with regards to safety and

operability of offshore structures for a long time. The green water generated by

extreme waves is primarily an air-water mixture, which includes a continuous phase

and a dispersed phase. This chapter reviews the general physics of green water

phenomenon and the developments of the BIV technique for measuring the dispersed

phase.

2.1 The green water phenomenon

Following BUCHNER [6], the main stages associated with the green water problem

are:

• Motions and relative wave motions

• Water flow onto the deck

• Water behaviour and loading on the deck

• Green water impact on structures

The relationship between the green water phenomenon and the incoming wave

is quite complicated. The complex fluid-structure interactions are still not well

understood. As commented by ERSDAL et al. [2], it is not necessarily the greatest

wave that causes maximum green water event. BUCHNER [21] and HAMOUDI

and VARYANI [22] also found that there is no direct relation between the velocity

of the incoming wave and the water velocity over the deck. STANSBERG et al. [23]

pointed that green water is generally the combined effect of two mechanisms, the

extreme wave events and negative pitch of the vessel. Hence, the pitch motion is

very important to the green water problem. In the present work, the influence of the

pitch motion is evaluated by a static pitch angle together with dam break model.

7

2.1.1 Scenarios of green water

GRECO et al. [24] suggested that the green water on deck is qualitatively associated

with the scenarios given in Fig. 2.1. Based on the ratio between the incoming wave

vertical velocity Ww and the vertical velocity W of water elevation at bow and in

terms of incoming wave steepness, the scenarios include the follows: dam break (DB,

figure a) type events characterized by flow of water along the deck similar to those

generated by the breaking of dams; initial plunging plus dam break (PDB, figure b)

type events, where the dam break water evolution is preceded by an initial plunging

phase; plunging wave (PW, figure c) type events, with the occurrence of a large-scale

plunging jet impacting on the deck and dominating the water-on-deck features; and

hammer fist (HF, figure d) type events. The ratio Ww

Wis used to characterize the

local effect of the ship on the free-surface elevation, while the wave steepness is a

measure of the incoming-wave non-linearities.

0

1

2

(a) (b)

(c) (d)

(e) ( f )

2

1

0

0 11

2

0

2

01 PDB Events

White Water/

HF Events

DB Events PW Events

DB Events

PW Events

PDB Events

White Water Events

HF Events

steepness

Figure 2.1: Scenarios of green water. Adapted from GRECO et al. [24]

The PDB type events are documented in GRECO et al. [7] and the HF type

events are documented in GRECO et al. [24]. According to BUCHNER [6], most

cases of green water flow onto the deck are resemble with theoretical dam break

model. Therefore, in the present work, we mainly consider the DB type events.

8

2.1.2 Green water impact on structures

In the green water problem, the horizontal velocity becomes dominant once the flow

reaches the deck level. When the high velocity water front on the deck reaches a

structure, this results in significant impact loading on the structure. The behavior

of the wave front resemble a jet impinging perpendicularly a vertical wall. In such

a scenario, BUCHNER [21] extended the slamming expression given by SUHARA

et al. [25] to the green water problem. In this expression the peak pressure depends

on the square of the velocity,

P = cρu2 (2.1)

where P is the peak impact pressure, c is called the impact coefficient, ρ is the fluid

density and u is the horizontal velocity. Based on empirical relations obtained from

experiments, SUHARA et al. [25] proposed an impact coefficient c = 1.4 for bot-

tom slamming situations. A different impact coefficient value of 0.88 was proposed

by BUCHNER [6], using Eq. (2.1) , and he commented that factor c = 1.4 was

conservative for the impact pressure.

BUCHNER [6] also noted a almost linear relationship between the impact force

and the maximum water height on deck, which is consistent with the experimental

results by FONSECA and SOARES [26]. BUCHNER [6] then introduced a new

expression as,

F = hmaxρu2 (2.2)

where F is the peak force per meter breadth, and hmax is the maximum water height

on deck.

Greco

Buchner

Figure 2.2: Schematic green water flow over the deck

GRECO [11] also followed the approach of representing the impact induced by

the green water problem as a slamming problem. She treated the problem as a

semi-infinite water wedge impact on a wall with wedge angle θ, shown in Fig. 2.2.

The zero-gravity similarity solution proposed by ZHANG et al. [27] was used to

study the influence on the impact coefficient, as shown in Fig. 2.3. It was found

9

that when θ ≤ 35◦ the peak pressure occurs at the intersection of the wall and the

deck (the bottom), while for larger angles the peak pressure shifts up. When the

angle is large enough (i.e θ > 60◦), the results agree qualitatively with the results

obtained using WAGNER [28] slamming theory.

5

00 20 40 60

10

(a) Peak impact pressure with different angle

5

10

00 5 10 15

(b) Pressure distribution along the verticalwall for 5 < θ < 75◦ with increment ∆θ =10◦

Figure 2.3: Results of similarity solution by ZHANG et al. [27], adapted fromGRECO [11]

The semi-infinite water wedge assumption by GRECO [11] and the similarity

solution by ZHANG et al. [27] were implemented in a numerical tool developed by

STANSBERG et al. [29, 30]. They concluded that the predicted force peak values

compared well with experimental data.

These researches indicates that the green water impact on structures are deter-

mined by the propagating wave front velocity and its shape (the angle θ). One may

use the classical “Stoker solution” to predict the wave front velocity and its shape.

By neglecting the bottom friction, “Stoker solution” predicts a constant wave front

velocity (2√gH) with a parabolic free surface profile and the tangent at the wave

front tip is horizontal. However, experimental data such as DRESSLER [31] and

LOBOVSKY et al. [32] have shown that both wave front velocity and its shape are

affected by the bottom friction. This emphasizes the necessity to include the bottom

friction in the analytical approaches.

More recent, ARIYARATHNE et al. [33] studied the green water phenomenon

for a breaking wave case. They suggested that the fluid density needs to be corrected

by introducing void fraction factor α for the bubbly multi-phase flow.

P = c(1− α)ρu2 (2.3)

For the wall impingement condition, the impact coefficient with void fraction factor

was found to be c = 1.3, which is close to the value of 1.4 reported in SUHARA

10

et al. [25].

2.1.3 Green water loading on deck

Once the green water flows onto the deck, it is not only impacts on the super-

structures, but also acts on the deck. By defining a control volume on the deck

and applying the Newton’s momentum relations, the total pressure on the deck was

given by BUCHNER [6, 21, 34] as:

P = ρ(g +∂w

∂t)h+ ρ(

∂h

∂t)w (2.4)

The first term represents the static pressure corrected for the vertical acceleration of

the deck, whereas the second term includes the effect of the rate of change of water

height on the deck.

OGAWA et al. [35] presented another approximation in regular waves,

F = αρgB(fe)2 (2.5)

where α is the coefficient, approximated as 0.3 by the results of experiment, ρ is

water density, g is gravity acceleration, B is the breadth of a ship, and fe is the

freeboard exceedance.

The green water loading on deck would contribute to the pitch motion of the

ship. In the present work, we simply the problem of the pitch motion with a static

pitch angle. Thus, the effects due to this loading would not be considered.

2.2 The BIV technique

Velocity measurement is one basic topic in fluid mechanics and various measurement

techniques have been developed.

Early quantitative measurement methods using Pitot tubes, Venturi tubes and

later measurement methods, such as Hot Wire Anemometry (HWA) and Laser-

Doppler Anemometry (LDA), by their nature, were measurement methods that

provided instantaneous velocity signals at single-points through time.

Recently, the advent of modern imaging, laser, and data acquisition technology

has allowed for the development and advancement of several relatively new mea-

surement techniques, namely Particle Tracking Velocimetry (PTV), Laser Speckle

Velocimetry (LSV), and Particle Image Velocimetry (PIV). These techniques allow

to measure the instantaneous velocity for a large flow field by recording images of

suspended seed particles in flows.

11

An important difference among the three techniques comes from the typical

seeding densities that can be dealt with by each technique. PTV is appropriate

with low seeding density experiments, PIV with medium seeding density and LSV

with high seeding density. In PIV, a typical interrogation region may contain images

of 10-20 particles. In LSV, the particle densities are so large that individual particles

are not distinguishable.

The Bubble Image Velocimetry (BIV) technique proposed by RYU et al. [15] is a

kind of direct adoption of the PIV technique. Hence, it implicit assumes the bubble

(seeding) density is medium. Issues may arise on this point, since RYU et al. [15]

claimed that BIV could be used to very high void fraction cases, which means the

bubble (seeding) density is high. In the present work, we assume a medium bubble

(seeding) density.

In the following section, the general description of the PIV technique and its

developments to the BIV technique is given.

2.2.1 General Description of PIV

Double

pulsed

laser

Light

sheet

opticsLight

sheet

Flow with

seeding particles

Measurement

volumeImaging

optics

CCD

Image frame

from pulse 1

Image frame

from pulse 2

Integration

area 1

Integration

area 2

Cross

Correlation

Figure 2.4: Typical arrangement for particle image velocimetry

A typical setup for PIV recording is shown in Fig. 2.4. Small seeding particles are

added to the flow. The seeded flow is illuminated by a thin light sheet generated by

a pulsed laser with its necessary optics. The sheet is pulsed twice and recorded by a

camera located perpendicular to the sheet. The recorded image pairs are processed

to determine the displacement of particles with a cross-correlation algorithm. The

displacements are then converted from the image pixelated domain to the spatial

12

domain via a calibration procedure. Finally, the particle displacements within the

spatial domain are then divided by the time separation between the laser pulses that

singly exposed sequential images, i.e. velocity = displacement/∆t, to provide the

velocity field.

The seeding particles added to the flow need to have control over their size,

density, and distribution. For ease of use, these particles should be nontoxic, non-

corrosive, and chemically inert. They should also be small enough to be good flow

tracers, yet large enough to scatter sufficient light for imaging. The density of the

particles should be as close as possible to the fluid. The distribution of particles

should be homogeneous. MELLING [36] presented a wide variety of tracer particles

that have been used in liquid and gas PIV experiments, as well as methods of

generating seeding particles and introducing them into the flow.

After the images containing particles are acquired, the most sensitive part of

a PIV analysis is the cross-correlation algorithm. The algorithm cross correlates

small sub images (interrogation areas) of an image pair to derive the most probable

particle displacement in the interrogation areas. Following HUANG et al. [37], the

cross-correlation for two discretely sampled images can be written as:

C(m,n) =∑i

∑j

A(i, j)B(i−m, j − n) (2.6)

where A(i, j) and B(i, j) are corresponding to the image intensity distribution in

the first and second image of a singly exposed image pair, m and n are the pixel

offset between the two images and C(m,n) is the cross-correlation function.

According to RAFFEL et al. [16], there are two common approaches to solve Eq.

(2.6). The most straightforward approach is to compute the correlation matrix in

the spatial domain. This approach is called direct cross correlation (DCC). Another

approach is to compute the correlation matrix in the frequency domain, called as

Discrete Fourier Transform (DFT). The DFT is calculated using a Fast Fourier

Transform (FFT). A graphical representation of the two cross-correlation algorithms

are shown in Fig. 2.5a and Fig. 2.5b respectively.

DCC has been shown to create more accurate results than DFT, according to

HUANG et al. [37]. However, regarding that several thousand velocity vectors can

be computed from one single pair of images the computational effort of DCC is

enormous. Given the size of a square interrogation area N , a number of operations

of the order of N4 have to be computed. On the other hand, the computational

effort of DFT is reduced to the O[N2 lnN ].

13

Shift (x=-1,y=2) Shift (x=2,y=2) Shift (x=0,y=0)

Shift (x=1,y=-2) Shift (x=-2,y=-1)

x

y Correlation Matrix

(a) Example of the formation of the correla-tion plane by DCC

(i,j)Real�to�complex

FFT

(i,j)

Complex�to�real

inverse FFT

Image 1

Image 2

at position (i,j)Image samplingInput

multiplicationconjugateComplex�

Real�to�complex

FFT

Output

Cross-

correlation

data

(b) Implementation of cross-correlation using FFT

Figure 2.5: The cross-correlation algorithms, adapted from RAFFEL et al. [16]

The most probable displacement of the particles from A to B is given by the

location of the intensity peak in the resulting correlation matrix C(m,n). By finding

the maximum value in C(m,n) and using a curve fitting technique for subpixel

accuracy, the mean particle displacement ( ∆x , ∆y) over the small area that occurs

at the highest correction can be obtained. The most common used curve fitting

technique is the three-point Gaussian fit, as shown in Fig. 2.6.

1 2 3 4 5 60

Corr

elat

ion m

atri

x i

nden

sity

Position (pix)

integer peak location sub-pixel peak location

Figure 2.6: Principle of the three-point Gaussian fit, adapted from THIELICKEand STAMHUIS [38]

14

The velocity can be subsequently calculated as,

u =∆x

∆t(2.7)

v =∆y

∆t(2.8)

2.2.2 Consideration for PIV in bubbly flow

In general, the classical PIV technique uses a thin laser light sheet for illumination

and with a perpendicular arrangement of the camera with respect to the light sheet.

For its application to bubble field, two main characteristics of the bubble should be

noted:

• The bubble size (typically in units mm) is even larger than the thickness of

thin laser sheet (typically several hundreds µm)

• The bubble light scattering is much different than the solid seeding particles,

for a spherical bubble, it may have reflection, refraction, second refraction, see

Fig. 2.8, and for non-spherical bubble, the light scattering would be even more

complicate.

Air Bubble

Figure 2.7: PIV image around an air bubble in a thin light-sheet, adapted fromBRUCKER [39]

15

Air Bubble

Optic Axis

incident light

Figure 2.8: Reflection and refraction of components of an incident ray for an airbubble, adapted from DAVIS [40]

For the first characteristic, as shown in Fig. 2.7 ,BRUCKER [39] commented

that as soon as the bubble enters the light-sheet, strong reflection occurs which

yields a shadow region behind the bubble. In the strong reflection area the light

intensity is too high (too bright) and in the shadow region the light intensity is

too low (too dark). These areas are lost for any valuable information. In addition,

ghost images seem to appear inside the bubble due to a mirror effect at the phase

boundary. These aspects strongly affect the quality of the results around the bubble.

For the second characteristic, BRODER and SOMMERFELD [41] found that the

scattering light intensity of air bubbles in water decrease strongly for off-axis angles

larger than 82.5◦. They concluded that the perpendicular arrangement of the camera

in the classical PIV technique was not suitable for taking images of bubbles in a

light sheet. Therefore, it is not considered to be suitable directly apply the PIV

technique for bubble velocity field measurement.

BRUCKER [39] pointed that to use the PIV technique for the dispersed phase, a

combination of the shadowgraph technique is recommended. A shadow image is cast

by bending of the light rays coming from the back side of the bubbles. The refraction

by strong curvature of the bubbles projects the dark part and hence casts a shadow

on the camera against the bright background. The early works of the combination

of PIV technique and the shadowgraph technique was refered to TOKUHIRO et al.

[42], DIAS and REITHMULLER [43], LINDKEN and MERZKIRCH [44],BRODER

and SOMMERFELD [45], FUJIWARA et al. [46], more recent, SATHE et al. [47].

As shown in Fig. 2.9, these studies were mainly focused on the low void fraction

cases (i.e single bubble rising) with the seeding particles. The bubble itself was not

treated as tracers and its velocity was evaluated by its surrounding seeding particles.

16

Bubble

around by

seeding

Shadowgraph

Figure 2.9: Sketch of a combination of the shadowgraph with PIV, adapted fromSATHE et al. [47]

2.2.3 General description of BIV technique

Recently, also based on the idea of combining the shadowgraph technique that il-

luminates the fluid from behind to reveal the flow pattern, and the classical PIV

technique that correlates the consecutive images to determine the velocity, RYU

et al. [15] introduced the so-called Bubble Image Velocimetry (BIV). The arrange-

ment of the BIV system is sketched in Fig. 2.10.

L

DOF

Light

High speed camera

Center of focal plane

Flume wall

Plastic glass

Light

Model

Figure 2.10: Sketch of BIV arrangements, adapted from RYU et al. [15]

17

In contrast to the previous PIV work in bubbly flow, RYU et al. [15] did not add

small seeding particles in the flow, but they treated the bubble itself as the tracer

particles. Since no seeding particles is added to the flow, the laser light sheet is

not required for the BIV technique. Instead, RYU et al. [15] used two regular 600

W light bulbs to illuminate the flow. The main light together with a translucent

plastic glass (as a diffuser) was placed at the back side of the tank to cast the

shadow images. The other light was placed on the front side to improve the light

intensity in the images. Since the BIV technique does not use a thin light sheet to

illuminate a specific plane of interest like the PIV method, it is necessary to know

where the measured bubbles are in the cross-tank direction. The measured volume

is determined by the depth of field (DOF) of the camera, which is given as,

DOF =Lf 2

f 2 − f#LC− Lf 2

f 2 + f#LC(2.9)

where L donates the distance between camera and the center of focal plane; f is

focal length of camera lens, f# is the camera aperture number and C represents

the circle of confusion. In the work of RYU et al. [15], L = 4.0 m, f = 105 mm,

f# = 1.8, and C = 0.03 mm, therefore the corresponding DOF is D = 0.15m.

Comparing to the PIV technique, in which the thickness of the thin light sheet is

typically 100 µm to 300 µm , the DOF is much larger. This becomes one of the

main error source of the BIV technique. RYU et al. [15] assumed this error due to

the thickness of the DOF in the obtained velocity can be estimated approximately

as,

ε ≈ DOF

2L= 1.88% (2.10)

RYU et al. [48] applied the BIV technique to study the runup and green water

velocity due to breaking wave impinging and overtopping. Based on the measured

data, they proposed a equation for predicting the horizontal velocity distribution of

green water along the deck. Based on the velocity obtained by the BIV technique,

RYU et al. [8] compared the green water velocity obtained by the BIV technique

with the theoretical “Stoker solution”, they concluded that the comparison was

surprisingly well. RYU and CHANG [49] employed the BIV technique together

with a fiber optic reflectometer (FOR) to measure the velocity and the void fraction

in the flow, and to determine the water level on the deck.

18

Chapter 3

Analytical Dam Break Solutions

Sudden destruction of a dam results in a highly unsteady flow, with a forward wave

advancing over a channel, and a backward disturbance wave propagating into the

still water. Dam break problem is one of the classical problems of unsteady open

channel flow.

This chapter deals with the dam break problem by the analytical approach and

is structured as follows. After this brief introduction, the theoretical background of

the two Stoker solutions is given, namely Eulerian Stoker solution and Lagrangian

Stoker solution. Following that, the attempts to include the bottom friction effects

are also described. Finally, a simplified dam break solution including the bottom

friction effects and bed slope effects is presented.

3.1 Stoker solutions

In this section we introduce the theoretical background for the two Stoker solutions.

Since ESS is the popular one, it would be described briefly while more details of

LSS would be given.

3.1.1 Eulerian Stoker solution

Under certain hypotheses, the dam break flow can be described by the Saint Venant

equations (SVE), which are written as follows in the 1-D sense:

∂h

∂t+∂hu

∂x= 0 (3.1)

∂u

∂t+ u

∂u

∂x+ g

∂h

∂x= g(S0 − Sf ) (3.2)

where h is the local depth at each section, u is the local horizontal velocity, S0 is the

bed slope and Sf is the friction slope. Eq. (3.1) comes from the continuity condition

across a channel section and Eq. (3.2) is the momentum equation. The basic

19

assumptions of SVE are that the flow is incompressible, the water depth is shallow,

the flow velocity is uniform over a cross section and the pressure is hydrostatic.

Considering a horizontal channel (S0 = 0) with a dry bed and ignoring the

bottom friction (Sf = 0), using the method of characteristics to solve the SVE,

STOKER [10] presented an analytical solution in Eulerian representation, the ESS,

which contains two expressions, Eq. (3.3) for the water depth and Eq. (3.4) for the

sectional velocity:

h =

H for

x

t< −√gH

1

9g(2√gH − x

t)2 for −

√gH <

x

t< 2√gH

0 forx

t> 2√gH

(3.3)

u =2

3(x

t+√gH) (3.4)

ESS is sketched in Fig. 3.1. ESS predicts a constant water level 4H9

at the dam

section x = 0 for time t > 0, which could be obtained by setting x = 0 in Eq.

(3.3). Since the bottom friction is ignored, the downstream wave front velocity is

also constant and equals to 2√gH.

Figure 3.1: Dam break in Eulerian representation, adapted from STOKER [10]

3.1.2 Lagrangian Stoker solution

An alternative solution to the dam break problem was also proposed by STOKER

[10]. This solution is based on the Lagrangian representation, as sketched in Fig.

3.2.

Assuming that the region occupied initially by the water is the half strip defined

by 0 ≤ a < ∞, 0 ≤ b ≤ H, where H denotes the height of the dam and a and b

represent Cartesian coordinates of the initial positions of the particles at time t = 0.

The dam gate is located at a = 0 and the displacements of the particles are denoted

by X(a, b; t), Y (a, b; t) and the pressure by p(a, b; t).

20

Figure 3.2: Dam break in Lagrangian representation, adapted from STOKER [10]

If gravity is assumed to be the only external force, then according to Newton’s

second law, the equations of motion are:

Xtt = −1

ρpX (3.5a)

Ytt = −1

ρpY − g (3.5b)

where subscripts indicate differentiation. From Eq. (3.5), STOKER [10] evaluated

the equations of motion in the Lagrangian form as:

XttXa + (Ytt + g)Ya +1

ρpa = 0 (3.6a)

XttXb + (Ytt + g)Yb +1

ρpb = 0 (3.6b)

where Eq. (3.6a) is obtained by multiplying Eq. (3.5a) with Xa and Eq. (3.5b)

with Ya, and add them; similarly, Eq. (3.6b) is obtained by multiplying Eq. (3.5a)

with Xb and Eq. (3.5b) with Yb, and add them.

For an incompressible fluid, the condition of continuity is expressed as follows:

XaYb −XbYa = 1 (3.7)

Following STOKER [10], we assume that X, Y and ρ can be expanded in power

series of time,

X(a, b; t) = a+X(1)(a, b)t+X(2)(a, b)t2 + ... (3.8a)

Y (a, b; t) = b+ Y (1)(a, b)t+ Y (2)(a, b)t2 + ... (3.8b)

p(a, b; t) = p(0) + p(1)(a, b)t+ p(2)(a, b)t2 + ... (3.8c)

where X(1) and Y (1) are the components of the initial velocity and X(2) and Y (2)

similarly for the acceleration. STOKER [10] stated that the series will converge for

sufficiently small values of the time. But the convergence characteristics of the series

21

have not been studied here. In order to determine the displacements terms (X(n)

and Y (n)) and the pressure terms (p(n)), the boundary value problem needs to be

solved.

Assuming the water is initially at rest, the boundary conditions of the displace-

ments follow:

X(a, b; 0) = a, Y (a, b; 0) = b (3.9)

Xt(a, b; 0) = 0, Yt(a, b; 0) = 0 (3.10)

The conditions (3.9) are automatically satisfied because of the form (3.8) chosen for

the series expansion. The conditions (3.10) are satisfied by taking X(1) = Y (1) = 0.

Initially, the free surface is represented by the particles on (0 ≤ a < ∞, b = H)

and (a = 0, 0 ≤ b ≤ H). When the dam is broken, the pressure can be prescribed

to be zero on the free surface:

p(a,H; t) = 0, 0 ≤ a <∞, t > 0 (3.11a)

p(0, b; t) = 0, 0 ≤ b < H, t > 0 (3.11b)

Inserting the power series (3.8a) in Eq. (3.6a) and considering the boundary condi-

tion on (0 ≤ a <∞, b = H), we have:

X(2)(a,H) = 0 (3.12)

again inserting the power series (3.8b) in Eq. (3.6b) and considering the boundary

condition on (a = 0, 0 ≤ b ≤ H), yields:

Y (2)(0, b) = −g2

(3.13)

Finally we consider the boundary conditions on the bottom (b = 0). Assuming

that the water particles originally at the bottom remain in contact with it, this

results:

Y (a, 0; t) = 0 0 ≤ a <∞, t > 0 (3.14)

From the condition (3.14), we have Y (2)(a, 0) = 0 (in fact, Y (n)(a, 0) would be zero

for all n).

By inserting the power series (3.8) in Eq. (3.6), we can build the relationship

between the pressure and displacement:

p(0)a = −2ρX(2) (3.15a)

p(0)b = −2ρY (2) − ρg (3.15b)

22

Considering the second derivative of pressure in a and b respectively, from Eq.(3.15)

we have:

p(0)aa + p

(0)bb = −2ρ(X(2)

a + Y(2)b ) = 0 (3.16)

Once p(0) is found, X(2) and Y (2) can be calculated and vice versa. Using the method

of conformal mapping, the solution to X(2) and Y (2) are given by STOKER [10] as:

X(2)(a, b) = − g

2πlog{

cos2 πb4H

+ sinh2 πa4H

sin2 πb4H

+ sinh2 πa4H

} (3.17a)

Y (2)(a, b) = − gπ

arctan{sin πb

2H

sinh πa2H

} (3.17b)

and then the solution to pressure p(0) is given as:

p(0) = ρg(H − b)− 8ρgH

π2

∞∑j=1

1

(2j − 1)2exp[−(2j − 1)π

2Ha] cos

(2j − 1)πb

2H(3.18)

The first term of p(0) represents the hydrostatic pressure.

For higher orders, STOKER [10] stated that a Poisson equation (similar to Eq.

(3.16)) can be found for p(n), with the right hand side determined by X(i) and Y (i)

for i = 2, 3, ..., n+ 2. STOKER [10] solved the Poisson equation for n = 0 and gave

the equation for n = 2 as:

∇2p(2) = − 8ρg2

H2(cosh πaH− cos πb

H)

(3.19)

YILMAZ et al. [50] solved the Eq. (3.19), which allows to give the solution to

X(4) and Y (4). However, the computing effort in X(4) and Y (4) is too much higher.

In the present work, we only use the LSS given by STOKER [10] and the shape

of the free surface profile is given by the following equations:

X = a+X(2)t2 (3.20a)

Y = b+ Y (2)t2 (3.20b)

evaluated for a = 0 (for the particles at the dam section) and for b = H on the

upper free surface.

23

3.2 Eulerian Stoker solution extensions

3.2.1 Bottom friction effects

In ESS, since the bottom friction effects are ignored, the dam break front velocity is

constant (U = 2√gH). However experimental data have shown that the wave front

velocity is not constant but a function of time.

To represent the bottom friction effects, DRESSLER [51] introduced the Chezy

resistance term K, proportional to u2

ghin the momentum equation, equivalently in the

form of friction slope Sf = K u2

gh. The velocity u and the depth h are then expanded

as power series in K, the first terms are the Stoker expressions. By finding the

next terms, he was able to see some of the modifications introduced by the bottom

friction. A convex shape of the wave front can then be predicted theoretically by

DRESSLER [51]. The validity of the results was latter confirmed by the same author

DRESSLER [31] with the following assumptions: (a) The product Kt must remain

small, (b) the velocity of the water does not vary rapidly in wave tip zone. With

the first order approximation, the front velocity is given by DRESSLER [31] as,

U√gH

= 2− 3.59(Kt

√g

H)13 + ... (3.21)

Later, WHITHAM [18] claimed that Dressler’s expressions would not be valid

near the wave front (called the tip region) where the frictional resistance is no

longer a mere correction term. To improve the solution, he treated the wave front

as a growing boundary-layer region and assumed that the bottom friction should be

only included in the tip region, differently from Dressler who included the resistance

in the whole domain. Based on experimental observation, Whitham also assumed

the front velocity is nearly uniform in the tip region.

StokerWave Tip Region

Figure 3.3: The Whitham solution

As shown in Fig. 3.3, x < x1 is the frictionless area where the ESS is valid. In

the tip region x1 < x < xf ,where the friction is important, the front velocity is

24

assumed to be uniform U . Thus, in the tip region, ut + uux is expected to be finite,

while hx becomes large, the momentum equation (Eq. (A.16)) is then reduced to,

hx +Ku2

gh= 0 (3.22)

By integrating Eq. (3.22), the wave profile in the tip region could be obtained,

h = U

√2K

g(xf − x) (3.23)

At x = x1, the velocity and the wave profile are continuous with the ESS,

x1 = (3U1

2−√gH)t (3.24a)

h1 = H(1− U1

2√gH

)2 (3.24b)

U1 = U (3.24c)

Based on momentum conservation in the tip region, note U = xf , yields,

(1− xf2√gH

)3 xf t√gH

= (1− xf2√gH

)4

− xf2

√gH R

H{ xf√

gH− (

3xf2√gH− 1)t}

(3.25)

Whitham solved this differential equation using a Taylor expansion. The first seven

terms are:

τ = 0.02431xf3 + 0.02163xf

4 + 0.01496xf5 + 0.00941xf

6

+ 0.00563xf7 + 0.00327xf

8 + 0.00186xf9 + ...

(3.26)

where, τ = Kt√

gH

. If only take the first term of this series, the front velocity is

given by,U√gH

= 2− 3.452(Kt

√g

H)13 + ... (3.27)

More recently, CHANSON [19] followed the first steps of Whitham’s work but used

mass conservation to obtain the front velocity, which led to a much simple for-

mulation for the front velocity. Another different point is that Chanson used the

Darcy-Weisbach friction factor f , with relationship K = f/8.

In the flow represented in Fig. 3.3, the conservation of mass must be satisfied.

Specifically the mass of fluid in the wave tip region (i.e. x1 ≤ x ≤ xf ) must equal

25

the mass of fluid in the ideal fluid flow profile for x1 ≤ x ≤ 2√gHt. It yields,

M =

∫ xf

x1

ρhWhithamdx =

∫ 2√gHt

x1

ρhStokerdx (3.28)

where, hWhitham could be obtained from Eq. (3.23) and hStoker could be obtained

from Eq. (3.3).

By integrating Eq. (3.28), it follows,

1

3

(1− U2√gH

)3

f8U2

gH

= t

√g

H(3.29)

3.2.2 Dam break wave in a sloping channel

The early studies attempt to extend the solution for a sloping channel are mainly

referred to HUNT [52] and CHANSON [19]. HUNT [52] presented a kinematic wave

equation for a downward sloping channel, this solution is available in CHANSON

[53], as shown in Fig. 3.4.

Wave front

H

L

U

Dam break wave

U

x

x

y

gate

Figure 3.4: Sketch of a dam break wave in a dry downward sloping channel, adaptedfrom CHANSON [53]

Hunt’s analysis gives:VHt

L=

1− (df/H)2

(df/H)3/2(3.30)

Xf

L=

3H

2df− ds

2H− 1 (3.31)

U

VH= −3VHt

4L+

√Xf + L

L+ (

3VHt

(4L))2 (3.32)

VH =

√8g

fHS0 (3.33)

where df is the dam break wave front thickness, Xf is the dam break wave front

position measured from the dam site, H is the reservoir height at dam site, L is the

reservoir length, S0 is the bed slope (S0 = H/L) and the velocity VH is the uniform

26

equilibrium flow velocity for a water depth H. From Fig. 3.4, it can be observed

that Hunt’s analysis assumes with a finite dam for the downward sloping bed case.

This is quite different to ESS which assumes a infinite dam.

CHANSON [19] also developed a solution for a frictionless dry upward sloping

channel, as shown in Fig. 3.5. This solution is called as CS, short for Chanson’s

solution. Chanson’s analysis is more similar to STOKER [10], using the method of

characteristics (Appendix A.1.2).

Figure 3.5: Sketch of a dam break wave in a dry upward sloping channel, adaptedfrom CHANSON [19]

Considering the characteristics curve C1, along C1, u + 2c − g(S0 − Sf )t is a

constant.

u+ 2c− gS0t = u0 + 2c0 − gS0t0 (3.34)

where subscript (0) donates the initial condition. Initially the water is at rest at

t0 = 0, this gives u0 = 0 and c0 =√gH. It yields,

u = −2c+ 2c0 + gS0t (3.35)

Considering characteristics C2 issuing from the dam break wave front,

dx

dt= u− c (3.36)

By substituting Eq. (3.35) in Eq. (3.36) and doing integration, it becomes,

x

t= −3c+ 2c0 +

1

2gS0t (3.37)

Noticing c =√gh, the water elevation could then be given as

h =1

9g(2c0 +

1

2gS0t−

x

t)2 (3.38)

Combining Eq. (3.35) and Eq. (3.37), the velocity formulation can be given as,

u =2

3(x

t+ c0 + gS0t) (3.39)

27

It is should be noted that the axis x and h are not orthogonal as shown in Fig. 3.5.

CHANSON [19] did not explain it directly but he claimed that these equations are

derived “for the initial stage of the dam break wave on a flat slope”. Thus whether

this could be worked for a slope bed case remains to be a question.

3.3 A piecewise solution

In this section, we introduce a piecewise solution to include the pitch motion (the

bed slope) for predicting shipping of water on deck. In reality, the relative motion

between the incoming wave and the ship pitch motion is quite complicated. Here,

we simplify the problem as shown in Fig. 3.6: the incoming wave always comes

horizontally and induces a horizontal dam; while the ship deck is varying due to

pitch motion and may induce both upward and downward sloping bed.

y

gate

Upward bed

Horizontal bed

Downward bed

downstream

x

upstream dam

frictionless part

wave tip

ESS

Figure 3.6: Sketch of a horizontal dam with different sloping bed

To deal with the changing in bed slope at x = 0, the problem is further simplified.

The whole domain can be divided into two part: the upstream and the downstream.

In the upstream side, the bed slope of the dam is horizontal, hence the upstream

part could be represented by the ESS: the wave elevation and velocity could be

predicted by Eq. (3.3) and Eq. (3.4) respectively. ESS predicts a constant water

level (4H9

) at the dam gate.

In the downstream side, we firstly consider the horizontal bed slope case. For a

horizontal bed case, we can follow WHITHAM [18] and CHANSON [19], as presented

in Section 3.2.1. Assuming the bottom friction is only included in the downstream

wave tip and the front velocity is uniform in the wave tip. The downstream side

could then be divided into two parts: the frictionless part (0 ≤ x ≤ x1) and the wave

tip part (x1 ≤ x ≤ xf ), as shown Fig. 3.6. The frictionless part for a horizontal

bed case can also be represented by ESS. In the wave tip, the wave front velocity

can be solved by Eq. (3.29) and the free surface profile is given by Eq. 3.23. Once

front velocity (U) is obtained, x1 and h1 can be calculated with Eq. (3.24a) and Eq.

(3.24b) using U1 = U .

28

Then for the sloping bed case, considering that the pitch angle is usually small

(less than 10 degree), we may assume the downstream elevation of a sloping bed

case (h(θ)) can be transformed from the elevation of the horizontal bed case (h) by:

h(θ) = −h sin(θ) + h (3.40)

Fig. 3.7 shows a example of the transformation from horizontal bed to upward

bed (θ = −10) and to downward slope bed (θ = 10). The benefit of this transfor-

mation lies that it does not require use of two coordinates system for upstream and

down stream.

x∗-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

h∗

-0.2

0

0.2

0.4

0.6

0.8

1

t∗ =0.66776, f =0.1

UpstreamInitial DamUpward Frictionless PartUpward Wave TipUpward Bed SlopeHorizontal Frictionless PartHorizontal Wave TipHorizontal Bed SlopeDownward Frictionless PartDownward Wave TipDownward Bed Slope

Figure 3.7: Example of transformation for θ = 10 and f = 0.1 at instant t∗ = 0.667

In summary, the piecewise solution (PS) consists of three part: upstream part,

downstream frictionless part and downstream wave tip part. The formulations to

determine elevation in each part are summarized in Table 3.1.

Table 3.1: Summary of the piecewise solution

Elevation Transformation

upstream h =1

9g(2√gH − x

t)2 No

frictionless part h =1

9g(2√gH − x

t)2 YES

wave tip h ≈√

f4U2

g(xf − x) YES

3.3.1 Solution with a compensated time factor

Introducing a compensated time factor tc in time as:

t′ = t− tc (3.41)

29

For instance, the compensated Stoker Euler solution (the upstream part of PS)

becomes:

h′ =

H for

x

t′< −√gH

1

9g(2√gH − x

t′)2 for −

√gH <

x

t′< 2√gH

0 forx

t′> 2√gH

(3.42)

u′ =2

3(x

t′+√gH) (3.43)

where superscript ′ donates the compensation with time. Considering time t′ could

not be negative, this means the compensation should be used at least when time

t > tc. The compensation in time will delay the velocity and improve the wave front

tracking ability in time t > tc. This would be discussed with experimental data in

the later part of this thesis.

30

Chapter 4

Numerical Dam Break Model

In this chapter the numerical code used for the simulation is described. The dam

break wave is a multiphase flow, including two immiscible fluids: air and water. To

simulate the evolution of the interface between two immiscible fluids two basic ap-

proaches can be followed: interface tracking method and interface capturing method.

The interface tracking method requires meshes that track the interfaces. The mesh

needs to be updated as the flow evolves. In the interface capturing method, the

computations are based on fixed spatial domains. In this study, simulations of the

dam break flow are based on the interface capturing method with a modified ver-

sion of the volume-of-fluid (VOF) approach, employed in the interFoam solver of

OpenFOAM version 2.3.0.

The present chapter is structured as follows. A brief overview of the OpenFOAM

toolbox is given first. The formulations of the interFoam solver would then be

followed. In the following sections, the modifications in the interFoam solver would

be described with extra attention. Finally, the interFoam solver is validated with

previously published experimental data.

4.1 Overview of OpenFOAM

OpenFOAM (Open Field Operation And Manipulation) is a free and open source

CFD toolbox. A general description about OpenFOAM could be found in JASAK

et al. [54] and JASAK [55]. Basically it consists a bundle of C++ libraries and

codes to solve complex problems such as turbulence, fluid flows, electromagnetics

and chemical reactions using finite volume discretisation. It also features several

applications to pre- and post-process the cases, including mesh generation tools

(blockMesh, snappyHexMesh), setting field values, mesh decomposition, sampling

data (isosurfaces, gauges). OpenFOAM is prepared to run cases in parallel, handling

itself the decomposition process and the final reconstruction process.

Unlike commercial codes the OpenFOAM is not a black box. By changing the

31

source code, every step of of the solving process is user controllable and modifiable.

This is a great advantage. It also includes several modules deal with data conversion

from and to commercial CFD codes (Ansys, Fluent, CFX). This allows the cross

validations with the commercial codes.

4.2 interFoam solver

The “interFoam” is one of the solvers included in OpenFOAM. It solves the three-

dimensional Reynolds Averaged NavierStokes (RANS) equations for two incompress-

ible phases using a finite volume discretisation and a modified volume of fluid (VOF)

method.

4.2.1 RANS equations

The RANS equations, which include equation of continuity (Equation 4.1) and equa-

tion of motion (Equation 4.2), are the governing mathematical expressions which

link pressure and velocity. The assumption of incompressible fluids has been used,

which is applicable for the dam break problem.

∇ ·U = 0 (4.1)

∂ρU

∂t+∇·(ρUU)−∇·(µeff∇U)−∇·(µeff dev(∇U)T ) = −∇prgh−g ·X∇ρ+σκ∇α

(4.2)

where ρ is the density; U is the velocity vector; prgh is the modified pressure; g

is the acceleration of gravity and X is the position vector. The terms include the

effective dynamic viscosity µeff are the stress terms, which are reformulated from

the deviatoric viscous stress tensor τ and the Reynolds stress tensor R to improve

the efficiency in numerical evaluation. The last term on the right is the effect of

surface tension: σ is the surface tension coefficient; κ is the curvature of the interface

and α is the water volume fraction, solved by the modified VOF method.

The elements in Equation 4.2 have a particular disposition: those placed on

the left hand side (LHS) of the equals sign are used in OpenFOAM to assemble

the coefficient matrix, and the ones on the right hand side (RHS) are calculated

explicitly and form the independent term of the equations.

The modified pressure, the reformulated viscous stress terms and the modified

VOF method would be described detailly in the later sections.

32

Modified Pressure

Considering the body force is only gravitational, to simplify the specification of the

pressure boundary conditions RUSCHE [56] introduced the modified pressure as:

prgh = p− ρg ·X (4.3)

The gradient of Equation 4.3 reads,

∇prgh = ∇p−∇(ρg ·X)

= ∇p− ρg − g ·X∇ρ(4.4)

RUSCHE [56] states that this treatment also enables an efficient numerical treat-

ment of the steep density jump at the interface by including the hydrostatic term

g ·X∇ρ into the RHIE and CHOW [57] interpolation. The Rhie-Chow interpolation

is the same as adding a pressure term, which is proportional to a third derivative

of the pressure. The object of Rhie-Chow interpolation is to eliminate pressure

oscillations and thereby also velocity oscillations. The oscillations occur if the pres-

sure gradient does not depend on the pressure in adjacent cells and thus allowing a

jigsaw pattern. Further discussion of the Rhie-Chow interpolation implemented in

OpenFOAM may refer to KARRHOLM [58].

Reformulated stress terms

The Navier-Stokes equations should be supplemented by an equation, which relates

the deformations and the stresses within the fluids. If the fluids obey the Newtonian

law of viscosity, the deviatoric viscous stress tensor is given by:

τ = µ(∇U + (∇U)T ) (4.5)

By applying the RANS approach, the Reynolds stress tensor is introduced,

R = ρU′U′ = −µt(∇U + (∇U)T ) (4.6)

where µt term represents the turbulent dynamic viscosity and is evaluated by the

chosen turbulence model. By defining the effective dynamic viscosity as,

µeff = µ+ µt (4.7)

33

the deviatoric viscous stress tensor and Reynolds stress tensor can be rearranged as,

∇ · (τ −R) = ∇ · [µeff (∇U + (∇U)T )]

= ∇ · (µeff∇U) +∇ · [µeff ((∇U)T − 1

3tr (∇U)T I +

1

3tr (∇U)T I)]

= ∇ · (µeff∇U) +∇ · [µeff (dev(∇U)T ]

(4.8)

where tr() means the trace of a matrix, defined to be the sum of the diagonal

components of a matrix, I represents the identity tensor (I ≡ δij ) and dev() donates

the deviatoric part of a matrix, i.e. for any matrix A, dev(A) can be evaluated as,

dev(A) = A− 1

3tr(A)I (4.9)

Note if the flow is assumed to be incompressible, the following relationship holds,

∇ ·U = tr (∇U)I = tr (∇U)T I = 0 (4.10)

Modified k − ε Model

The nonlinear Reynolds stress term requires additional turbulence model to close

the RANS equation. OpenFOAM supports several turbulence models, e.g. Spalart-

Allmaras model, k−ε model and k−w model. The most common turbulence model,

the k − ε model, is used for this study. The k − ε model implemented in interFoam

solver is a modified version, it reads,

∂ε

∂t+∇ · (Uε)−∇ · (( νt

σε+ ν)∇ε) = Cε1

ε

kG− Cε2

ε

kε (4.11)

∂k

∂t+∇ · (Uk)−∇ · (( νt

σk+ ν)∇k) = G− ε

kk (4.12)

where G is the production rate of kinetic energy due to the gradients in the resolved

velocity field,

G = 2νt|1

2(∇U + (∇U)T )|2 (4.13)

Comparing to the standard k − ε model of LAUNDER and SPALDING [59], the

density is not included in Equation 4.11 and Equation 4.12 but taking into account

with the turbulent viscosity,

µt = ρνt = ρCµk2

ε(4.14)

34

The model constants are,

Cε1 = 1.44 Cε2 = 1.92 Cµ = 0.09 σk = 1.0 σε = 1.3 (4.15)

4.2.2 Modified VOF method

One of the critical issues in numerical simulations using the VOF method is the

sharp resolution of the interface while preserving the boundedness and conservation

of the phase fraction. This is especially in cases of flow with high density ratios such

as air and water in the dam break wave, where small errors in volume fraction may

lead to significant errors in calculations of physical properties. To overcome these

difficulties, the VOF method was modified with the so-called surface compression

approach. In the interFoam solver, the α-equation reads,

∂α

∂t+∇ · (Uα) +∇ · [Urα(1− α)] = 0 (4.16)

where the phase fraction α is the indicator function defined in Equation 4.17 and

Ur is an extra, artificial compression term introduced by RUSCHE [56], defined as

the relative velocity between phase a and b (Equation 4.18).

α =

0, for points belonging to one phase;

0 < α < 1, for points at the phase-interface;

1, for points belonging to another phase.

(4.17)

Ur = Ua −Ub (4.18)

Using the indicator function, two immiscible fluids are considered as one effective

fluid throughout the domain, the local density ρ and the local viscosity µ of the fluid

are given by:

ρ = αρa + (1− α)ρb (4.19)

µ = αµa + (1− α)µb (4.20)

where the subscripts a and b denote the different fluids.

The role of the artificial term∇·[Urα(1−α)] is to compress the interface. Because

of the multiplication term α(1 − α), the artificial term contributes only within the

interface region and vanishes at both limits of the phase fraction. Therefore, it does

not affect the solution significantly outside this region. As commented by RUSCHE

35

[56], the main advantage of such formulation lies in the higher interface resolution

comparison with the conventional VOF approach. Numerical diffusion, unavoidably

introduced through the discretization of convective terms, can be controlled and

minimized through the discretization of the compression term, thus allowing sharp

interface resolution. The other advantage is the boundedness, the solution of α

is bounded between zero and one while the original VOF method of HIRT and

NICHOLS [60] does not preserve local boundedness on the phase fraction.

Calculation of surface tension

Surface tension is a tensile force tangential to the interface separating the two fluids

which tries to keep the fluid molecules at the free boundary in contact with the rest

of the fluid. Accurate calculation of the phase fraction distribution is crucial for

a proper evaluation of surface curvature, which is required for the determination

of surface tension force and the corresponding pressure gradient across the free

surface. To evaluate the surface tension force, the continuum surface force (CSF)

model proposed by BRACKBILL et al. [61] is implemented. The CSF model is

suitable for fixed Eulerian meshes and applies a volumetric force in those cells of

the mesh containing the transitional interface region as an approximation to the

discontinuous surface tension force. The surface tension force, appearing in the

momentum expression(Equation 4.2) , is represented as

fσ = σκ∇α (4.21)

where the mean curvature of the free surface κ is computed from local gradients in

the surface normal at the interface,

κ = −∇ · ( ∇α|∇α|

) (4.22)

4.2.3 Discretised model equations

The finite volume discretization in OpenFOAM of the temporal terms and spatial

terms (convection term, diffusion term and source term) are described in detail

by JASAK [62]. The solution domain is subdivided into a number of cells with

computational points placed at the cell centroids. Each two cells share exactly one

cell face, on which a surface normal vector Sf is defined, as shown in Figure 4.1.

36

P

Nd

Figure 4.1: Discretization of the solution domain, adapted from GREENSHIELDS[63]

With Gausss theorem applied to the convective and diffusive terms and using

the Euler implicit time scheme 1, the following discretized system of equations is

derived, ∑f

Uf · Sf = 0 (4.23)

ρnUn − ρoUo

∆tVP +

∑f

ρfφUnf −

∑f

µefff Sf · ∇fUn −

∑f

µefff (dev(∇Uo)T )f · Sf =∑f

[∇fprgh − gXf∇fρ+ (σκ)f∇fα]|Sf |

(4.24)

αn − αo

∆tVP +

∑f

αfφ+∑f

φrαf (1− α)f = 0 (4.25)

where VP represents the control volume, subscript ()f implies the value of the vari-

able in the middle of the face, superscript ()n denotes the current (new) time step,

superscript ()o denotes the previous (old) time step and operator ∇f denotes a

discretized gradient at the face,

∇fUn = (∇Un)f =

UnN −Un

P

|d|(4.26)

where subscript ()P indicates the value in the current cell and subscript ()N indicates

the value in the neighbour cell, and φ is surface velocity flux,

φ = Uf · Sf (4.27)

1This time scheme is used for the demonstration of the discretization process; for the simulationslater on, the CrankNicolson time scheme is used.

37

4.2.4 Pressure-velocity coupling

The pressure-velocity coupling is solved with the so-called PIMPLE algorithm. PIM-

PLE is a merged algorithm of Pressure Implicit Splitting Operator (PISO) and

Semi-Implicit Method for Pressure-Linked Equations (SIMPLE). Its main structure

is inherited from the original PISO, but it allows equation under-relaxation to en-

sure the convergence of all the equations at each time step. A detailed description

of the SIMPLE and PISO algorithm can be found in FERZIGER and PERIC [64]

and ISSA [65], respectively.

Momentum predictor

The left hand side (LHS) of Equation 4.24 is assembled and called UEqn in the

interFoam solver.

UEqn:ρnUn − ρoUo

∆tVP +

∑f

ρfφUnf−∑f

µefff ·∇fUn−∑f

µefff (dev(∇Uo)T )f ·Sf

(4.28)

The UEqn could be assembled in solution vector form as AUn = b, with A

being the momentum coefficient matrix, Un the unknown velocity vectors and b the

vector containing the source terms. The following operators acting on the UEqn

can be defined and are implemented in OpenFOAM: the D-operator AD gives the

diagonal part of matrix A, the N-operator AN represents the off-diagonal part of

matrix A, the S-operator AS extracts the source vector b. With these operators,

the UEqn becomes,

(AD + AN)Un = AS (4.29)

Equation 4.24 can then be rewritten in a semi-discretized form,

(AD + AN)Un = AS −∇porgh − gXf∇ρ+ σκ∇α (4.30)

Since the exact pressure gradient is not known, the old pressure filed (porgh) from

the previous step is used to predict the velocity Un. It should be noted that do-

ing a momentum predictor is not an essential step for the convergence of the PISO

pressure corrector loop although it is sometimes benefitial but not always. There-

fore in interFoam solver the momentum predictor is optional by setting indicator

momentumPredictor to yes or no.

38

Pressure corrector

The first corrected velocity U∗ is being solved from the predicted velocity Un and

the first corrected pressure p∗rgh.

ADU∗ + ANUn = AS −∇p∗rgh − gXf∇ρ+ σκ∇α (4.31)

The problem is that the corrected pressure is yet unknown - all that known is the

old pressure. To obtain the corrected pressure, introducing,

AH = AS −ANUn (4.32)

By inverting AD, Equation 4.31 becomes,

U∗ = AD−1AH −AD

−1∇p∗rgh + AD−1(−gXf∇ρ+ σκ∇α) (4.33)

This will be used later to calculate the face fluxes. By replacing Equation 4.33 in the

continuity equation, the Pressure Poisson Equation of the first corrected pressure

p∗rgh can be written as:

∇ · (AD−1∇p∗rgh) = ∇ · (AD

−1AH + AD−1(−gXf∇ρ+ σκ∇α)) (4.34)

Further pressure correction steps can be applied using the same AD matrix and

AH vector. This is convenient computationally since they can be stored in mem-

ory and recalled as necessary. It should be noted that although it is possible to

recalculate the coefficients in AH after each pressure solution, it is not done in in-

terFoam. According to JASAK [62], the non-linear coupling is less important than

the pressure-velocity coupling. The coefficients in AH are therefore kept constant

through the whole correction sequence and will be changed only in the next momen-

tum predictor.

Extra flux correction term

It is not guaranteed that the implementation of the solvers in any CFD code, in-

cluding OpenFOAM, is faithful to the original theory behind the algorithms. As in

all CFD codes, the implementation may have some extra artificial terms to improve

some features such as stability robustness. In commercial solvers such modifications

are kept in the black box and cannot be tracked but in open source codes the code

are free for evaluation. In the following we briefly discuss one special flux correction

term in the implementation of PISO algorithm.

Taking a further look at the solver code, it reveals that in the pressure correction

39

loop of time step n the mass flux is of the form

φnPISO = φ+ φc (4.35)

The first term is the standard velocity flux defined in Equation 4.27 and the second

term φc is the extra correction term. This extra term is specific to OpenFOAM and it

is not a part of the original PISO implementation of ISSA [65]. The implementation

of φc in OpenFOAM is formulated as,

φc =Kc

∆t(AD

−1)f [φoPISO − (U o)f · Sf ] (4.36)

where the correction coefficient (0 ≤ Kc ≤ 1) is evaluated as follows,

Kc = 1−min(|φoPISO − (U o)f · Sf ||φoPISO|+ ε

, 1) (4.37)

where ε is a very small number to avoid division by zero.

It seems φc may affect the divergence of the face velocity since the interpolated

velocity becomes subtracted from the flux. However, to the best of our knowledge,

the influence of this extra flux correction term is not documented yet and requires

further investigation.

4.3 Validation of the solver

As mentioned in the previous sections, the interFoam solver include many specific

modifications. To validate the solver, a dam break case is set up with results com-

pared to experimental data available from LOBOVSKY et al. [32].

The computation domain is shown in Fig. 4.2 based on the setup of LOBOVSKY

et al. [32]. The length of the upstream reservoir between the dam gate and the left

wall equals 600 mm and the downstream channel length is 1010 mm. Four pressure

sensors P1 to P4 are located in the right wall, with different heights 3, 15, 30 and

80 mm.

40

6001000

H WATER

1610

Atmosphere

P2

P3

P4

Lower wall

Rig

ht

wal

l

Lef

t w

all

30

80

15 3

P1

Figure 4.2: Schematic of dam break model,in units: mm, adapted from LOBOVSKYet al. [32]

The initial condition is set to be zero for all velocity components and the modified

pressure. The boundary is defined in two groups: the walls and the atmosphere. The

no-slip boundary condition is specified for all the walls ; the boundary conditions

for the turbulence kinetic energy k and its dissipation rate ε are specified with the

aid of the technique of wall functions. For the atmosphere, the zero values for the

pressure, water volume fraction and all velocity components are also specified.

The OpenFOAM utility blockMesh is used to discretize the computation domain

as hexahedral blocks. For the simulation a uniform mesh with Ds = ∆x = ∆y = ∆z

is adopted. The feature of “adjustable run-time” available in interFoam is used with

the maximum CFL number set to be 0.25.

The discretization schemes used in the interFoam solver is shown in Table 4.1.

The “Gauss” keyword specifies the standard finite volume discretization of Gaussian

integration. TVD stands for total variation diminishing scheme and CD is short for

central differencing scheme. The symbol ∗ stands for an arbitrary scalar or vector.

41

Table 4.1: The discretization schemes of interFoam

Category Term Discretization schemes Note

Temporal ∂∂t∗ Euler Euler Implict

Divergence

∇ · (ρUU) Gauss linearUpwind Linear upwind differencing

∇ · (µeff dev(∇U)T ) Gauss linear Linear interpolation (CD)

∇ · (Uα) Gauss vanLeer TVD van Leer limiter

∇ · [Urα(1− α)] Gauss linear CD

Laplacian

∇ · (µeff∇U)

Gauss linear corrected

CD

∇ · (( νtσε

+ ν)∇ε) with Explicit correction

∇ · (( νtσk

+ ν)∇k) on non-orthogonal meshes

Gradient ∇∗ Gauss linear CD

Surface Nor-

mal Gradient(∇∗)f corrected

Explicit correction on

non-orthogonal meshes

The simulation results of case H = 600 mm is shown in Figure 4.3. To evaluate

the influence of the mesh size, three different mesh size DS = 20, 10, 5 mm is used.

It is observed from the comparison with the experimental data (the solid line) in

Figure 4.3, the agreement of the pressure the signal for all pressure sensors P1 to

P4 is good.

42

t√

gH

0 1 2 3 4 5 6 7 8

P/(ρ

g H

)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5Sensor P1

Lobovsky 600mmDS=20 mmDS=10 mmDS=5 mm

1.4 1.6 1.81.5

2

2.5

3

3.5

t√

gH

0 1 2 3 4 5 6 7 8

P/(ρ

g H

)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5Sensor P2


1.4 1.6 1.81.5

2

2.5

3

3.5

t√

gH

0 1 2 3 4 5 6 7 8

P/(ρ

g H

)

-0.5

0

0.5

1

1.5

2

2.5Sensor P3


1.4 1.6 1.81.5

2

2.5

3

3.5

t√

gH

0 1 2 3 4 5 6 7 8

P/(ρ

g H

)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Sensor P4


1.6 1.8 21

1.2

1.4

1.6

Figure 4.3: Results of pressure comparing to experimental data

4.3.1 Grid convergence

According to ROACHE [66, 67], the grid-convergence index (GCI) presents a simple

method for uniform reporting of grid-convergence studies. The GCI is based on the

Richardson Extrapolation (RICHARDSON and GAUNT [68]), which is a method for

obtaining a higher-order estimate of the continuum value (value at zero grid spacing)

from a series of lower-order discrete values (f1, f2, ..., fn). Using the grid refinement

ratio r, ROACHE [66] generalized Richardson extrapolation by introducing the pth-

order methods:

fexact ≈ f1 + (f1 − f2)(rp − 1) (4.38)

and the order-of-accuracy can be estimated by using the following equation:

p =ln(ε32)/ ln(ε21)

ln(r)(4.39)

εi+1,i = fi+1 − fi (4.40)

where fi+1 is a coarse-grid numerical solution and fi is a fine-grid numerical solution.

Then the GCI for the fine grid solution can be written as:

GCIi+1,i = Fs ∗ | εi+1,i

1− rp| (4.41)

43

where Fs is the safety factor, following ROACHE [66], the safety factor selected for

this study is 3.

According to STERN et al. [69], the possible convergence conditions of the sys-

tem are: 1) monotonic convergence (0 < R < 1); 2) oscillatory convergence (R < 0);

3) divergence(R > 1). And R is the convergence ratio and it is determined by the

equation:

R =ε21

ε32

(4.42)

The order of accuracy and the GCI for the impact peak pressure for all pressure

sensors (p1, p2, p3 and p4) from the simulation results of three grids (DS = 20, 10, 5

mm) are summarized in Table 4.2.

Table 4.2: Order of accuracy and Grid Convergence Index for three integrationvariables. Subscripts 3, 2 and 1 represent case DS = 20, 10, 5 mm, respectively.

ε32 ε21 R p GCI32 GCI21

p1 0.634 0.565 0.891 0.166 15.574 13.879

p2 0.765 0.339 0.443 1.174 1.826 0.809

p3 0.036 0.020 0.556 0.848 0.135 0.075

p4 0.068 0.054 0.794 0.332 0.787 0.624

As listed in Table 4.2, there is a reduction GCI value for the successive grid

refinements (GCI21 < GCI32). The GCI for finer grid (GCI21) is relatively low

if compared to the coarser grid (GCI32), indicating that the dependency of the

numerical simulation on the cell size has been reduced. Additionally, the convergence

ratios R for all pressure sensors (p1 to p4) are less than one, according to STERN

et al. [69], the convergence conditions are monotonic.

44

Chapter 5

Dam Break Experiments

This chapter is structured as follows. The general experiment setup would be given

first. All devices used for data acquisition would be described in detail later. Finally,

the test cases are included.

5.1 Experiment setup

Oqus 310

Oqus 110

6-DOF Platform

Water

ReservoirBed

Blocks

Weight

with

Electromagnet

Gate

LED 2

Figure 5.1: Experimental setup, overall view

The experiment setup was built and installed in the LabOceano facility. Fig. 5.1

shows an overall view of whole setup. It consists of an acrylic tank divided into two

separate parts by a removable gate and a gate release system. The tank is installed

on the 6-DOF platform available in the laboratory which guarantees a horizontal

tank bottom. The detailed setup is shown in Fig. 5.2, including all recording

equipments. All parts of the setup are further explained hereafter.

45

Figure 5.2: Experimental setup of the dam-break tests

5.2 Tank and gate release system

The tank is made of 15 mm thick acrylic with inner dimensions of 780× 335× 600

mm. The angular deviation of the bottom of tank is 0.12 degree and 0.15 degree

in the x and y directions respectively. A 10 mm thick acrylic dam gate is placed

385 mm from the lateral side of the tank. This defined the dam reservoir region

(upstream) and the channel region (downstream).

One challenge of dam break experiment is the water-tightness of the dam gate.

The dam gate should be watertight but also be removable. In this study, the

Ethylene-vinyl acetate (EVA) foam is used at the bottom side and lateral side of the

gate. To decrease the friction , the lateral side of dam gate is smeared with silicone

grease.

As shown in Fig. 5.3, the gate release mechanism consists of two fixed pul-

leys, an electromagnet and a 10 kg releasing mass with a damping reservoir half

filled with sand. This arrangement was similar to the release mechanism used by

STANSBY et al. [12] and LOBOVSKY et al. [32]. To control the gate releasing, an

electromagnet was used and controlled by a global TTL trigger signal. Once the

global TTL trigger signal was set to logic level LOW, the 10 kg mass was dropped

under gravity releasing the gate and starting video acquisition. The distance be-

tween the electromagnet and the damping reservoir was 1.1 m. The weight falls

freely a distance of 0.6 m, reaching a velocity about 2-3 m/s. This arrangement

46

is to achieve the so-called ”sudden dam break”, as demonstrated in LAUBER and

HAGER [17]: the nondimensional gate removal period is smaller than t√

gH

=√

2,

with g = gravitational acceleration, H = initial water depth and t = time.

Sand

Electromagnet 10 kg

6 DOF Platform

about 1.1 m

0.6 m

0.65 m

0.6 m

Figure 5.3: Schematic view of the gate release system

5.3 Data acquisition and synchronization

The data acquisition flow chart is shown in Fig. 5.4. A global TTL trigger signal is

used to set the time origin (t = 0). It allows all measured variables with synchronized

records related to the same time base.

Wave

Probes DHI Wave

Amplifier 102 E

NI SCXI 1314

Terminal Block A

Pressure

sensors

Temperature

sensors

TTL

Triger signalNI PCIe 6361

PC for

Data StorageNI SCXI 1520

Module

NI SCXI 1000

Chassis

Camera

Oqus 310

Camera

Oqus 110

PC for

Data Storage

Electromagnet

NI SCXI 1314

Terminal Block B

Synchronization

Gate Release

Mechanism

NI SCB 68

Connector Block

NI SCXI 1314

Terminal Block B

NI SCXI 1520

Module

NI SCXI 1520

Module

NI SCXI 1001

Chassis

Accelerometer

NI PCIe 6361

PC for

Data Storage

NI SCB 68

Connector Block

Sample Rate

32 KHz

Sample Rate

60 Hz

Figure 5.4: Data acquisition Flow

Considering the requirements in capturing different sensor signals, e.g. pressure

sensors and wave probes, are different, two different sample rate are used. The

47

higher sample rate of 32 KHz is applied for pressure sensors and accelerometer and

the lower sample rate of 60 Hz is applied for wave probes and temperature sensors.

The choice of higher sample rate is based on the pressure nature frequency, which

would be described in the Section 5.3.2.

For the video recording, the global TTL trigger signal is set as the external

trigger signal of the Qualisys Track Manager (QTM) software. The synchronization

between camera Oqus 310 and camera Oqus 110 is controlled by the QTM software

with the internal time clock.

5.3.1 Wave probes

The wave probe comprises two thin, parallel 316L stainless steel electrodes. When

immersed in water, the wave probes measure the conductivity of the instantaneous

water volume between the two electrodes, a conductivity that changes proportion-

ally to changes of the water surface elevation, i.e. the wave height, between the

electrodes.

The conductance, G, between two parallel electrodes is determined as:

G = Cwlπ

ln 2Dd

(5.1)

where Cw is the conductivity of the water, l donates the length of immersed part of

the electrodes, d is the diameter of the electrodes and D is the distance between the

electrodes (center to center). In the present work, d = 2.3 mm and D = 8.0 mm.

The DHI Wave Amplifier 102E is used for data acquisition. To outbalance the

influence of temperature or salinity changes of the water, the compensation is set

manually inside the DHI Wave Amplifier based on the water conductivity. The

water conductivity is 850 µS/cm, measured by the TDS&EC conductivity meter

with an accuracy of 2%.

The wave probe locations are specified in Fig. 5.2. This arrangement make it

possible to do a comparison of the new data with previously published researches,

such as BUCHNER [6], LEE et al. [70] and LOBOVSKY et al. [32].

In the upstream, two wave probes (h7, h8) are used to measure the water el-

evation. Probes are mounted in pair outside the center line to observe the three-

dimensional effects in the flow. In the downstream, initially six wave probes (h1

to h6) were used. However, the Reynolds number of the downstream flow reaches

Re = 2√gHDν≈ 6.7E3, with water depth H = 0.2 m and wave probe diameter D =

2.4 mm. The Reynolds number is so high that unwanted turbulence wake vortex is

found as shown in Fig. 5.5.

48

Figure 5.5: Downsteam turbulence

One may use smaller wave probes or change the water viscosity to reduce the

Reynolds number. Considering the limited resources available in laboratory, we

overcome this difficulty in the other way, measuring the water elevation indirectly.

T

hree thin black plastic markers are used in the downstream instead of the real

conductive wave probe. They are called as virtual wave probes (h1 , h3 and h5).

The time history of water free surface elevation is evaluated by imaging analysis

from captured video images. The algorithm of elevation detection is included in

Appendix C.1.

(a) front view (b) top view

Figure 5.6: Wave probes and virtual wave probes

5.3.2 Pressure sensor

At the downstream side, five pressure sensors are used to measure the impact pres-

sure. Sensor p1, p2, p3 and p4 are located in the centerline while sensor p5 is

49

located outside the center-line but with the same height level of sensor p2, as shown

in Fig. 5.2. This arrangement allows to observe the three-dimensional effects. The

sensor positions are set near the bottom side of the tank right wall, because the

peak pressure occurs at the bottom side.

The requirement for the pressure sensors is special. The sensors need to be as

small as possible, operate in a large range of pressure magnitudes, possess a good

resolution and be capable of measuring in two phases flows. There are various types

of pressure-sensing technologies, such as piezoresistive, capacitive, electromagnetic,

piezoelectric, optical, and potentiometric types. For impact pressure of green water

or sloshing phenomenon, piezoresistive and piezoelectric type of sensor are mainly

applied. The instruments available in laboratory are the EPX Miniature pressure

sensor produced by Measurement Specialties Inc. It is a piezoresistive type pressure

sensor, with a range of 7 bar and a diameter of the head of 5 mm. The specifications

are shown in Table 5.1. Following SOUTO-IGLESIAS et al. [71], with our experi-

mental capabilities, the bias uncertainty in the pressure measurements is about 10

milibar.

Table 5.1: Specifications of EPX sensor

Description Value Units

Pressure range 0-7 bar

Resonant frequency 80 KHz

Thermal Zero Shift ±1.5% FSO / 50 ◦C

Sensing area diameter 5 mm

Operating Temperature -40 to 120 ◦C

It should be noted that the useful frequency (the frequency response limit) of the

sensor is about 20 % of the resonant frequency, that is 16 KHz. According to the

Nyquist Shannon sampling theorem, the sampling rate should be chosen as twice

of the highest frequency component, that is 32 KHz. The pressure sensors should

be mounted carefully to avoid anomalous behavior that could be attributed to the

mounting. On one hand the so-called flush mount is required which means that

the membrane surface of the pressure sensor is as coincidence as possible with the

surface of the test object. On the other hand, the installation torque might affect

also. Excessive torque may change zero offset. In such cases, output should be

re-zeroed through instrumentation.

50

5.3.3 Temperature sensors

To evaluate the temperature shock effect for the pressure sensors, two LM35 Pre-

cision Centigrade Temperature Sensor T IN and T OUT are used. As shown in

Fig. 5.2, sensor T IN is located inside the tank under the water to measure the

water temperature and sensor T OUT is located outside the tank to measure the

environment air temperature.

The LM35 series are precision integrated-circuit temperature devices with an

output voltage linearly proportional to the Centigrade temperature. The LM35

device does not require any external calibration or trimming to provide typical

accuracy of ±0.25 ◦C at room temperature.

5.3.4 Accelerometer

Acceleration during the test may also affect the quality of the results. For instance,

vibration induced by the friction between the dam gate and tank wall may affect

the pressure measurement. To evaluate the possible acceleration effects, one MEMS

accelerometer Model 4803A is installed on the downstream side wall.

The Model 4803A is an tri-axial accelerometer in a rugged, welded stainless steel

package. It incorporates integral temperature compensation that provides a stable

output over a wide temperature range from -55◦ to +125◦. The acceleration range

is ±2g.

5.3.5 Video recording and illumination

Two high-speed digital cameras Oqus 310 and Oqus 110 are used for observing the

dam break flow motion. The Qualisys Track Manager (QTM) software is used for

the acquisition of the high-speed videos. The field of view of camera Oqus 310 is

mainly on the downstream side to capture more details of the dispersed phase of

the dam break flow. Camera Oqus 110 is set at the top side of the tank to observe

the three dimensional effects of the flow. The camera properties are shown in Table

5.2.

51

Table 5.2: Camera properties

Camera Type Oqus 310 Oqus 110

Resolution 1280 × 1024 640 × 480

Aperture Number 2.0 1.5

Focal Length (mm) 105 8.5

Sensor Type CMOS CMOS

Sensor Size (mm) 6.3 × 4.7 17.9 × 14.3

Max FPS 502 247

Used FPS 500 200

For illumination, the so-called shadowgraphy technique is applied. The LED

panel with dimension 500×400 mm is chosen as the light source due to the following

two advantages: firstly comparing to AC powered 600 W light bulbs used in RYU

et al. [15] or the halogen lights used in CHENG et al. [72], the DC powered LED

panel does not have the light flicker effect and have less light fluctuation; secondly

the light density distribution of the LED panel is much more uniform. To improve

the light density uniformity, a light diffuser is set at the back side of the tank, as

shown in Fig. 5.2.

5.4 Test cases

To study the effects of the bed slopes, five different slope bed blocks were built and

would be placed at the bottom of the tank, as shown in Fig. 5.7 and Table 5.3. The

height of downstream side for all the blocks are the same, equal to 98 mm, while

the height of the upstream side varies for different angles.

52

394.50 384.50

Front View

Top View

334

Figure 5.7: Sketch of the slope bed blocks

Table 5.3: Slope bed blocks and test cases

Block Name LU (mm) LD (mm) θ (degree) H (mm)

Flat 98 98 0 110, 220

5A 132 98 5 110, 220

5B 64 98 -5 110, 220

10A 166 98 10 110, 220

10B 30 98 -10 110, 220

For each block, two different water level cases are tested, including a lower water

level case with H = 110 mm and a higher water level case with H = 220 mm.

Therefore, totally ten cases are tested. Each case is repeated ten times.

53

Chapter 6

Results of Dam Break Models

After the gate releasing, the dam break wave is advancing on the channel and later

impacting on the wall and reflecting backward. The whole dam break period can be

divided into three main different periods, named as: the main period, the impact

period, and the reflection period.

The main period lasts until the impact happens. Since the analytical dam break

solutions work only in the main period, this period is in the mainly focus. In this

period, we evaluate the dam break wave free surface profile by comparing the results

of analytical solutions, numerical simulations, and experimental data. The “sudden

dam break” concept is also investigated.

After the main period, the downstream dam break wave is impacting on the

tank wall and running up along the wall, named as the impact period. In the

impact period, the pressure is the main topic. Finally, a reflection wave is generated

and propagated backward, called as the reflection period. This period is less relevant

to the green water problem.

6.1 Main period with Stoker solutions

In the present section, we compare the free surface profile for the horizontal bed case

of H = 110 mm. The results of both analytical solutions and numerical results are

plotted over the experimental images as shown in Fig. 6.1, where the solid line is the

ESS, the dotted line represents the LSS, the dots are the simulation results obtained

with OpenFOAM and the dashed line indicates the initial dam. The OpenFOAM

simulation results are not extracted from a cut plane but the projection of all points

in the X-Z plane, as shown in Fig. 6.2a and Fig. 6.2b. The coordinate system is

transformed from the world (in units mm) to image (in units pixel) based on the

camera calibration data.

54

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

Figure 6.1: H=110 mm, evaluation of free surface profile

55

0.8

0.6

x: m

0.4

0.2

00

0.05

0.1

0.15

y: m

0.2

0.25

0.3

0.02

0.04

0.06

0.18

0.16

0.14

0.12

0.1

0.08

0.35

z: m

(a) free surface profile in 3D

x: m0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

z: m

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

(b) free surface profile projected in X-Zplane

Figure 6.2: OpenFOAM free surface profile

In general, the comparison between numerical results and experimental data

shows a very good agreement on the free surface profile. On the other hand, based

on agreement between the analytical solutions and the experimental data, the main

period can be divided into three stages. Theses stages are named as: the gate

releasing stage (Fig. 6.1a to Fig. 6.1c), the transition stage (Fig. 6.1d to Fig. 6.1f)

and the later stage (Fig. 6.1g to Fig. 6.1j).

The first gate releasing stage is defined from the initial instant (t = 0) until the

instant when the gate displacement is equal to the initial water level (H). At this

position, the gate has no more influence in the flow. We name this instant as tg.

This interval can be associated with the gate removal period defined by LAUBER

and HAGER [17]. They proposed that this period in non-dimensional form should

be smaller than√

2 for the flow to be considered as a sudden dam break.

From Fig. 6.1c, this instant can be estimated as tg = 50 ms. In dimensionless

form (t∗g = tg√g/H), it corresponds to 0.47, which is almost one third of

√2.

Following LAUBER and HAGER [17], the present case (H = 110 mm) can be

considered as a “sudden dam break”.

Comparing the analytical solutions to the experimental data, it is possible to

observe that the LSS agrees quite well in both upstream and downstream side of

the gate. On the other hand, the ESS overestimates the position of the wave front

on both sides of the dam gate.

Following the gate releasing stage is the transition stage, defined from the end

of the releasing stage (tg) to the instant when ESS and LSS have an intersection

point at the dam gate section. We name this instant as t1. The intersection point

can be estimated from the ESS. As mentioned in Section 3.1.1, the water level at

the dam section is constant and equals to4H

9. Therefore the intersection point of

the two solutions occurs at (0,4

9H). Knowing where the intersection occurs we can

56

find now the instant t1.

P1

P2

P1

P2

t=0

t=t1

a

a

b

0

b=H

b=0

0

b

a=0

Figure 6.3: Sketch of the initial stage of the dam break problem with top particleP1 and bottom particle P2 at the dam gate section

In the Lagrangian representation, the dam can be represented by the points P1

and P2 shown in Fig. 6.3. Initially, both particles are at rest. The position of P1

is defined as X(0, H; 0) = 0 and Y (0, H; 0) = H; while for P2 is X(0, 0; 0) = 0

and Y (0, 0; 0) = 0. From the bottom boundary condition (b = 0) described in

Section 3.1.2, particle P2 would move only in the horizontal direction. On the other

hand, with the boundary conditions at the dam section (a = 0), we have X(2)P1

= 0

and Y(2)P1

= −g2, which indicates that particle P1 would move only in the vertical

direction. At instant t = t1, particle P1 reaches to position (0, 4H9

). By representing

the YP1 in power series, we have:

YP1 = H − g

2t1

2 =4H

9(6.1)

Solving Eq. (6.1), we obtain the instant t1 equals to√

103

√Hg

and in dimensionless

form t∗1 =√

103

. For the case of H = 110 mm, t1 corresponds to 111.7 ms. This value

is almost twice of the gate removal period. From Fig. 6.1f, we can observe that the

ESS and LSS cross the intersection point at the instant 112 ms. This agrees with

our results quite well.

In the transition stage, the agreement with experimental data of LSS is still bet-

ter than ESS in the upstream side; while for the downstream side, both solutions

have a poor agreement. It is observed that ESS overestimates the wave front lo-

cation; while for LSS, one may observe the wave front location is underestimated

57

but with many particles (represented by dots) propagating far away on the bottom.

This nonphysical behavior seems to be related to the limitations in LSS. Firstly,

by inspecting Eq. (3.17a), there is a logarithmic singularity for particle P2 (a = 0,

b = 0) in X(2). Secondly, since Y (2) is always negative, the displacement Y would be

negative as time t increases. Although with the bottom boundary condition (3.14),

the particles are forced to contact with the bottom.

The last stage, named as the “later stage”, occurs for time greater than t1. In the

later stage, it is observed that as time increases, the curve of LSS falls down rapidly

and finally almost vanished. On the other hand, as time increases, the agreement of

ESS with experimental data gets much better except the wave front region where the

bottom friction can not be ignored anymore. However, the bottom friction effects of

the dam break problem are out of the scope of the present paper, for the reader who

has interest in this topic is referred to the work of DRESSLER [51] and WHITHAM

[18].

Physically, if we consider particle P1, it may be explained as follows. Initially,

by neglecting the friction of the dam gate, particle P1 behaves as free fall under the

influence of gravity. The free fall velocity of particle P1 is v(t) = gt and the free fall

distance s(t) = 12gt2. At time t = t1, the fall distance s(t1) = 5H

9. This gravitational

potential energy would be changed into kinetic energy in both horizontal direction

and vertical direction. One may find in ESS, only the horizontal velocity component

is considered. On the other hand, in LSS, due to the boundary condition X(2)P1

= 0,

the horizontal velocity component of P1 is zero, only the vertical velocity component

is given. Both solutions have its own limitations. As can be observed, during the gate

releasing stage and the transition stage (t < t1), the vertical movement is dominant.

In this period, the free fall assumption seems to be valid and the free surface profile

is better described with LSS. While in the late stage (t > t1), the flow propagates

mainly in the horizontal direction. In this period, the free fall assumption is not

valid and the free surface profile is better represented by ESS.

The behavior observed could be an important information to develop a more

general solution to the dam break problem in the future, combining the LSS and

the ESS. Thus, it is important to verify that our findings are not restricted to the

present case (H = 110 mm). The experimental data available from LOBOVSKY

et al. [32] is then used. Similarly, we identified the three stages for the case of

H = 300 mm. The results are shown in Fig. 6.4, where tg = 100 ms (Fig. 6.4b) and

t1 ≈ 183 ms (Fig. 6.4c). In dimensionless form, the gate removal period t∗g equals

0.57, which is less than half of the value (√

2) proposed by LAUBER and HAGER

[17]. Comparing to the present case (H = 110 mm), the same observations in the

three stages can be made.

58

(a) (b)

(c) (d)

Figure 6.4: H=300 mm, evaluation of free surface profile, adapted from fromLOBOVSKY et al. [32]

6.1.1 Sudden dam break

Based on the findings presented in the previous section, it is important to review

the definition of “sudden dam break”. As can be observed, the dimensionless gate

removal period of√

2 proposed by LAUBER and HAGER [17] is much larger than

the value in our case (t∗g = 0.47) for H = 110 mm and in the case (t∗g = 0.47) of

LOBOVSKY et al. [32] for H = 300 mm. It seems the value of√

2 overestimates

the gate removal period.

Note that LAUBER and HAGER [17] obtained this value√

2 with the following

assumption: the top particle P1 behaves as free fall until P1 reaches to bottom. As

discussed before, until time t = t1, particle P1 falls to the location 4H9

and after this

instant, P1 would move mainly in the horizontal direction. The free fall assumption

would not be valid in the later stage. This leads us to revise “sudden dam break”

gate removal period proposed by LAUBER and HAGER [17].

Whether a dam break is “sudden” or not is directly related to the gate releasing

velocity. The average gate releasing velocity could be estimated as

Vg =H

tg(6.2)

and in dimensionless form V ∗g = Vg√gH

. Clearly, V ∗g is a function of the initial water

level H. To study the influence of H on V ∗g , four different cases are taking into

59

consideration. The first two cases are done in the present work with a lower water

level (H = 110 mm) and a higher water level (H = 220 mm); the other two cases

are available from the work of LOBOVSKY et al. [32], also with a lower water level

(H = 300 mm) and a higher water level (H = 600 mm).

(a) H = 220 mm, present case (b) H = 600 mm, from LOBOVSKYet al. [32]

Figure 6.5: Results of the two higher level cases at instant tg

Table 6.1: The average gate releasing velocity and the gate removal period

H (mm) tg (ms) Vg (m/s) t∗g V ∗g Agreement with LSS

Present case 110 50± 2 2.20 0.47 2.12 GOOD

Present case 220 100± 2 2.20 0.67 1.50 BAD

LOBOVSKY et al. [32] 300 100± 3.3 3.00 0.57 1.75 GOOD

LOBOVSKY et al. [32] 600 216.7± 3.3 2.77 0.87 1.14 BAD

The results of the two lower level cases at time tg have already been shown in

Fig. 6.1c and in Fig. 6.4b respectively. It is observed that for the two lower level

cases, the agreement of free surface profile with LSS at time tg is good. On the other

hand, for the two higher level cases, as can be observed from Fig. 6.5, the agreement

of free surface profile is poor that the curve the LSS is below the water free surface.

This observation with the calculations of Vg and tg for all four cases are summarized

in Table 6.1, where tg is obtained from the video frames. The uncertainty of tg could

be estimated as 1FPS

, where FPS donates frame per second. In the present work

FPS = 500 and in the work of LOBOVSKY et al. [32], FPS = 300.

This may be explained as follows. In the present work, since the gate releasing

system remain the same for different water levels, then Vg for both water level cases

are almost equal. However, the dimensionless velocity V ∗g = Vg√gH

is decreased for

60

the higher water level case, as can be observed in Table 6.1. This argument is the

same for the work of LOBOVSKY et al. [32]. In the two higher water level cases,

as V ∗g is decreased, the dam gate is not released ”suddenly” enough. Due to the

existence of the gate, the discharge rate of the dam is reduced. Therefore at time tg

the dam holds more water than predicted by LSS. Thus, the two higher level cases

could not consider as a “sudden dam break”. It should be noted that even for the

two higher level cases, t∗g is still smaller than the value of√

2 proposed by LAUBER

and HAGER [17]. This confirms that the value of√

2 proposed by LAUBER and

HAGER [17] is overestimated.

From the data in Table 6.1, it could be found that the critical value of t∗g might

be within the range of [0.57, 0.67], which is much smaller than the value proposed

by LAUBER and HAGER [17]. Further studies are required to find this exact value.

In the present paper, as a first estimation, we recommend that the “sudden dam

break” occurs when the gate removal period is smaller than3t∗15

= 3√

105≈ 0.63.

6.2 Main period with the piecewise solution

6.2.1 Horizontal bed cases

In the previous section, the main period is evaluated with the two Stoker solutions,

in which bottom friction is not taking into account. The tangent of the free surface

profile at the wave front tip predicted by ESS is horizontal, which does agree with

a convex shape as can be observed in experimental, for example, in Fig. 6.1.

In this section, we evaluate the piecewise solution (PS) presented in Section 3.3,

which takes account the bottom friction and bed slopes. Firstly, we consider the

horizontal bed case. The results for the present case H = 220 mm and for case

H = 600 mm available from LOBOVSKY et al. [32] are shown in Fig. 6.6a and Fig.

6.6b respectively.

(a) H=220 mm, present case, withf=0.12

(b) H=600 mm, adapted fromLOBOVSKY et al. [32], with f=0.16

Figure 6.6: Example results of the piecewise solution in horizontal channel

61

For a horizontal bed case, the PS actually is an extension of ESS by taking into

account the bottom friction in the wave tip (wave front). The shape of the wave

tip predicted by PS seems close to the experimental data, however, the wave tip

location predicted by PS seems to be overestimated.

To improve the tracking ability of the wave front, we introduce a time compen-

sation factor tc in the PS as:

t′ = t− tc (6.3)

The idea of tc comes from the findings described in the previous section that ESS

only works fine in the later stage. Thus, in the view of using ESS, the early times is

not important and we may consider to compensate it. By taking t∗c =t∗13

, for present

cases H = 110 mm and H = 220 mm, the results are shown in Fig. 6.7 and Fig.

6.8 respectively.

Figure 6.7: H=110 mm, results of the PS with compensated time t∗c =t∗13

, and withbottom friction factor f = 0.04

62

Figure 6.8: H=220 mm, results of the PS with compensated time t∗c =t∗13

, and withbottom friction factor f = 0.12

It can be observed that the agreement between the piecewise solution with the

experimental data becomes quite well in both main part and wave tip area. On one

hand, the wave front location is well tracked by introducing the time compensation

factor. On the other hand, the convex shape of the wave tip (wave front) is also well

captured by choosing the bottom friction factor as f = 0.04 for case H = 110 mm

and f = 0.12 for case H = 220 mm. Comparing to Fig. 6.1, the agreement with

experimental data for the piecewise solution is much better than the ESS.

To verify that the piecewise solution with the time compensation are not re-

stricted to the present cases, we also applied it to one previously published case H

= 600 mm, available from LOBOVSKY et al. [32]. The results are shown in Fig.

6.9. A well agreement is also observed, by choosing f = 0.16. These observations

indicate that by using the compensated time factor as t∗c =t∗13

and a proper bottom

friction factor according to experiment, the PS now can track the wave front not

63

only in its advancing location but also in its convex shape in the wave tip.

Figure 6.9: H=600 mm, adapted from LOBOVSKY et al. [32], results of the PS

with compensated time t∗c =t∗13

, and with bottom friction factor f = 0.16

Wave front angle estimation

As shown in Fig. 2.3, the water wedge angle is an important parameter to estimate

the Green Water peak impact pressure. Assuming a horizontal bed case, since the

wave front now can tracked by the compensated PS, it is possible to estimate the

wedge angle θ as:

θ = arctan∆h

∆x(6.4)

where ∆h is the maximum height of the wave tip and ∆x is wave tip length. They

can be estimated from the wave tip part solution of the compensated PS, as shown

64

in Section 3.3.1 and the wedge angle is given by,

θ = arctanh′1

x′f − x′1= arctan

2f8

U ′2FgH

(1− U ′F2√gH

)2(6.5)

6.2.2 Slope bed cases

The slope bed cases can also be evaluated by PS with a compensated time factor

t∗c =t∗13

. The results for the different bed slope bed cases are summarized in Table

6.2.

Table 6.2: Summary of the different bed slope bed cases

Slope Angle (degree) H (mm) f Figure

Downward 10110 0.04 Fig.6.10

220 0.12 Fig.6.11

Downward 5110 0.04 Fig.6.12

220 0.12 Fig.6.13

Upward 10110 0.04 Fig.6.14

220 0.12 Fig.6.15

Upward 5110 0.04 Fig.6.16

220 0.12 Fig.6.17

The following observations could be made:

• From Fig.6.10 to Fig.6.17, for all bed slope cases, for time large than t∗ ≈t∗c + 0.3 ≈ 0.63 ≈ 3t∗1

5, the downstream free surface profile between the PS

and experimental data agrees relatively well. The wave front location and its

shape is well captured by using a constant bottom friction factor (f = 0.04

for cases H = 110 mm and f = 0.12 for cases H = 220 mm).

• PS assumes a constant water level 4H9

at the dam section. This assumption

seems to be valid for the downward slope cases (Fig.6.10 to Fig.6.13); while

the upward slope bed cases, the water level at the dam section seems to be

larger than 4H9

, as shown in Fig.6.14 to Fig.6.17.

• The free surface profile between the numerical simulation results by Open-

FOAM and the experimental data agrees well.

As described previously, the time interval within [0,3t∗15

] might be assumed to

be the gate releasing stage for a sudden dam break. Since t∗c is introduced in the

65

solution, the solution would not work for t∗ < t∗c . We recommend to use PS with tc

for time large than3t∗15

. For time less than3t∗15

, it relates to the initial stages of dam

break problem where the time compensation factor should not be used.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 6.10: H=110 mm, evaluation of free surface profile for downward slope angle10 degree, with t∗c =

t∗13

and with f = 0.04

66

(a) (b)

(c) (d)

(e) (f)

(g) (h)


t∗13

and with f = 0.12

67

(a) (b)

(c) (d)

(e) (f)

(g) (h)


t∗13

and with f = 0.04

68

(a) (b)

(c) (d)

(e) (f)

(g) (h)


t∗13

and with f = 0.12

69

(a) (b)

(c) (d)

(e) (f)

(g) (h)


t∗13

and with f = 0.04

70

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 6.15: H=220 mm, evaluation of free surface profile for upward slope angle10 degree, with t∗c =

t∗13

and with f = 0.12

71

(a) (b)

(c) (d)

(e) (f)

(g) (h)


t∗13

and with f = 0.04

72

(a) (b)

(c) (d)

(e) (f)

(g) (h)


t∗13

and with f = 0.12

6.2.3 Bottom friction factor

Questions may arise on the bottom friction factor. The bottom friction factor is

related to flow resistance. The flow resistance is quite a big topic, as pointed by

YEN [73]:“Over the years there have been numerous investigators who have made

important contributions to open-channel flow resistance. ”

73

According to ROUSE [74], the flow resistance could be classified into four compo-

nents: 1) surface or skin friction, 2) form resistance or drag, 3) wave resistance from

free surface distortion, and 4) resistance associated with local acceleration or flow

unsteadiness. By using the Darcy-Weisbach resistance coefficient f , he expressed

the resistance as the following dimensionless symbolic function:

f = F (Re,K, η,N, Fr, Un) (6.6)

in which Re = Reynolds number; K = relative roughness, usually expressed as

ks/RH , where ks is the equivalent wall surface roughness and RH is hydraulic radius

of the flow; η = cross-sectional geometric shape; N = nonuniformity of the channel;

Fr = Froude number; Un = degree of flow unsteadiness; and F represents a function.

For a special case of steady uniform flow in straight constant diameter rigid pipes,

ROUSE [74] evaluated with Eq. (6.6) yields to the commonly called the Moody

diagram. There are other empirical formulations to estimate the flow resistance for

steady flow. Two different approaches are showed in the following sections.

Approach A

According to CHANSON [53], the bottom friction may be estimated with the Chezy

coefficient (C) and the Darcy-Weisbach coefficients (f) as,

C =1

nR

16H =

√8g

f=

√g

K(6.7)

where RH is the hydraulic radius, calculated by the tank width W and initial water

level H:

RH =W ∗HW + 2H

(6.8)

and n is the Manning roughness coefficient. According to CHANSON [53], the

Manning roughness coefficient for the plastic or glass can be estimated as 0.01. The

bottom friction factor are calculated in Table 6.3.

Table 6.3: Bottom friction factor calculation with Approach A

Cases W (m) RH (m) C f

Present H = 110 mm 0.335 0.066 63.63 0.019

Present H = 220 mm 0.335 0.095 67.56 0.017

LOBOVSKY et al. [32] H = 300 mm 0.15 0.060 62.57 0.020

LOBOVSKY et al. [32] H = 600 mm 0.15 0.067 63.67 0.019

74

Approach B

CHANSON [53] and CHANSON [75] also proposed another empirical formulation

(called Altsul formula) as:

f = 0.1(1.46ksDH

+100

Re)14 (6.9)

where ks is the typical roughness height and DH is the hydraulic diameter and Re

is the Reynolds number defined as:

Re =

√gHDH

ν(6.10)

Table 6.4: Bottom friction factor calculation with Approach B

Cases DH (m) Re f

Present H = 110 mm 0.266 3.07E+05 0.0576

Present H = 220 mm 0.380 4.40E+05 0.0527

LOBOVSKY et al. [32] H = 300 mm 0.240 4.59E+05 0.059

LOBOVSKY et al. [32] H = 600 mm 0.267 6.49E+05 0.057

Notes on bottom friction factors

The calculated bottom friction factors in Table 6.3 and Table 6.4 with different

approaches are different. Both approaches shows that the bottom friction factors is

only slightly affected by the initial water level H.

To the best knowledge of the author, these empirical formulations are suitable

for the steady flow. However, the dam break wave is an unsteady flow. According

to ROUSE [74], the unsteadiness of the dam break flow contributes to the flow

resistance. These empirical formulation seems not to be suitable for the unsteady

dam break flow. As shown in Table 6.2, the bottom friction factor is quite different

for H = 110 mm and H = 220 mm. The agreement of these empirical formulations

does not seem to be good. Due to the limited time, the bottom friction factor would

be left for further researches.

75

6.3 Pressure measurement

6.3.1 Typical signal of pressure

A typical signal of the pressure sensors and the zoom pressure peaks are shown in

Fig. 6.18. The following aspects should be paid attention to:

1. small peaks before impact: due to gate releasing

2. pressure drop after impact: related to phase changing effects

3. the ability of the pressure Return-To-Zero (RTZ) after impact

Time: s1 2 3 4 5 6 7 8 9

Pre

ssure

: m

Bar)

-1000

-800

-600

-400

-200

0

200Pressure

p1p2p3p5

(a) A typical signal of pressure sensors

Time: s1.072 1.074 1.076 1.078 1.08 1.082 1.084 1.086 1.088

Pre

ssure

: m

Bar)

-10

0

10

20

30

40

50

60

70

80

90

Pressure

p1p2p3p5

(b) Zoomed pressure peaks

Figure 6.18: A typical signal of pressure sensors and the zoomed pressure peaks

Small peaks due to gate releasing

During the gate releasing stage, due to the friction between the gate and tank, there

is vibration on the structure. This vibration leads to the strong variation in the

acceleration signal and damping quickly. The vibration yields small variations on

the pressure signal during the gate releasing: when the gate start releasing and when

the gate reach the top, as shown in Fig. 6.19. Fortunately, before the impact peak

pressure, those small variations on pressure have already damped almost to zero.

Thus the vibration related to gate releasing seems not affect the measurement of

the impact peak pressure.

Pressure drop related to phase changing effects

To study the sudden pressure drop as shown in Fig. 6.18a, by taking advantage of the

6 DOF platform, the periodical pitch motion is applied to evaluate the performance

of the pressure sensors. The setup of the pitch motion test is shown in Fig. 6.20, no

76

Time: s0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Pre

ssure

: m

Bar)

-1000

-800

-600

-400

-200

0

200Pressure

p1p2p3p5

time: s0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Accele

ration: m

/s2)

-40

-30

-20

-10

0

10

20

30

AccX

AccY

AccZ

start release

gate reach top

Figure 6.19: Small peaks before impact related to the gate releasing

Front View Side View

p2p1

p2,p5

p4p3

p4p3

p5

BASE of 6-DOF Platform

Acc.

Acc.

p1

100 mm

pitch motion

Figure 6.20: Pitch motion test with 6 DOF platform

bed blocks is used. The still water level is set to 100 mm to make sure all pressure

sensors initially out of water.

Fig. 6.21 shows the results of pitch motion test in period of 3 seconds and 10

seconds, with the same amplitude of the pitch angle in 10 degree. In period of of

10 seconds, the platform moves slowly, a wave was generated in the tank, pressure

sensors p1, p2 and p5 was initially dry but later are always underwater due to the

wave; pressure sensors p3 and p4 are in a higher position, for this two sensors, there

is a phase changing from air to water due to the wave periodically. In period of 3

seconds, the platform moves faster, the amplitude of the generated wave is higher

that the phase changing from air to water is happened for all pressure sensors.

Consistently a sudden pressure drop is observed: in period of 10 seconds for sensor

p3 and p4; in period of 3 seconds for all sensors.

From the above observation, it could be concluded that the pressure drop is re-

lated to the phase changing effects. This phase changing may relate to the thermal

77

time:s0 5 10 15 20 25 30

Acc:

m/s

2

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

ACC_YACC_ZACC_X

time:s0 5 10 15 20 25 30

pre

ssu

re:

mB

ar

-400

-300

-200

-100

0

100

200

300

400Pitch

p1p2p3p4p5

(a) Period of 3 seconds

time:s0 10 20 30 40 50 60

Acc:

m/s

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

ACC_YACC_ZACC_X

time:s0 10 20 30 40 50 60

pre

ssu

re:

mB

ar

-200

-150

-100

-50

0

50

100

150

200

250

300Pitch

p1p2p3p4p5

(b) Period of 10 seconds

Figure 6.21: Pitch motion test in different periods, the amplitude of the pitch angleis 10 degree

shock effects as called by VASSILAKOS et al. [76] or temperature shock effects as

called by PISTANI and THIAGARAJAN [77]. In the convenience for later discus-

sion, it is simply called as the thermal shock effects. As demonstrated by PISTANI

and THIAGARAJAN [77], the thermal shock effects are caused by the sudden cool-

ing of the dry sensor when it comes in contact with the liquid. The sudden cooling,

due to the contact with the liquid, causes a non homogeneous contraction on the

pressure sensor sensing surface that apparently results in an unwanted deformation

in the element underneath. VASSILAKOS et al. [76] also studied the thermal shock

effects and concluded that since heat transfer is involved, the thermal response takes

a large fraction of a second to become evident, thus the thermal shock effects are

78

not important as they occur much later in time than the initial pressure pulse. This

means that the thermal shock effects do not necessarily imply inaccuracy of the

pressure readings during the impact event, but it could be a concern for the later

time pressure response.

However, this pressure drop maybe more complex than the thermal effects.

Firstly, the temperature for water and air are measured as 24.6◦C and 25.3◦C re-

spectively to two LM35 Precision Centigrade Temperature Sensor T IN (for water)

and T OUT (for air). Thus this difference in temperature of air and water is quite

small, less than one degree. Secondly, it is observed from Fig. 6.21 that the phase

changing velocity seems also has its contribution on the pressure drop. Initially all

sensors are dry and out of water, when the platform starts to move, the movement

is slow and the pressure sensor contact with liquid gently. For both periods, there

are a small initial pressure drop. This small pressure drop is more related to the

temperature difference between air and water, since the phase changing velocity is

slow. While for the later stages, the velocity of the generated wave increased, with

the increased phase changing velocity, the periodical pressure drop is much larger.

What’s more, considering the initial pressure drop for both periods, the initial pres-

sure drop in period of 10 seconds is lower comparing to the period of 3 seconds. It

is also could be explained as the phase changing velocity is lower for period of 10

seconds than for period of 3 seconds.

Although this pressure drop maybe a important aspect for the pressure mea-

surement, but how to overcome it has been out of the scope of this study. It is

recommended further studies on this topic.

Return-To-Zero ability

Significant concern also exists regarding the ability of the pressure transducers to

return to zero reading following a water impact event. Fig. 6.18a shows the pressure

signal after the pressure drop (due to the thermal shock effects) remain negative and

do not return to zero. This maybe related to the properties of the sensor itself. The

pressure sensors used in the work are produced by the Measurement Specialties Inc.

(short as MS). VASSILAKOS et al. [76] studied this issue with different pressure

sensors, one from MS also, the other is from Endevco Corp. The Endevco sensors

are found with RTZ ability while the MS sensors do not have the RTZ ability.

Unfortunately, the pressure sensors available for this work are MS sensors and they

do not have the RTZ ability.

79

Repetitions

t ∗

√

g

H

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

p

ρgH

-2

-1

0

1

2

3

4

5

6

7pressure

p1_cases01p1_cases02p1_cases03p1_cases04p1_cases05p1_cases06p1_cases07p1_cases08p1_cases09p1_cases10Mean Peak

(a) p1

t ∗

√

g

H

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

p

ρgH

-2

-1

0

1

2

3

4

5

6

7

8

pressure


(b) p2

t ∗

√

g

H

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

p

ρgH

-2

-1

0

1

2

3

4

5

6

7

8

pressure


(c) p3

t ∗

√

g

H

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

p

ρgH

-2

-1

0

1

2

3

4

5

6

pressure


(d) p5

Figure 6.22: Pressure peaks in ten times repetitions for horizontal bed case H = 110mm

Due to the existence of the pressure drop effects, the pressure signal after of the

peak is not in interest. The important is the impact peak pressure. The zoomed

pressure peak is shown in Fig. 6.18b. The peak signals of sensor p1, p2 and p3 are

in time sequence according to the position order from low to high. The difference

in sensor p2 and p5 may implies the possible 3 D effects.

Fig. 6.22 shows the pressure peaks in ten times repetitions, where the mean

peak value is marked with the red dashed line. Comparing to LOBOVSKY et al.

[32] (the mean peak value is about 3.2), the mean peak value measured in this work

is larger.

80

Notes on pressure sensor

Although a lot of time has been spent to improve the pressure signal quality, unfor-

tunately with the limited resources in both instrumentation and time, the quality

of the pressure signal captured is not considered to be good, due to the following

aspects: (a) high noise to signal ratio ( in Fig. 6.22, the variation before the peak

signal may upto 20% of the peak value ); (b) significant pressure drop due to phase

changing effect; (c) lacking of RTZ ability.

6.3.2 Numerical pressure results

The numerical simulation results are shown in Fig. 6.24 for case H = 110 mm and

Fig. 6.24 for case H = 220 mm. It is expected that for downward slope bed cases

with a higher pressure and for upward slope bed cases with a lower pressure.

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.5

0

0.5

1

1.5

2

2.5

3

H=220 mm, p1

Downward 10 degreeDownward 5 degreeHorizontalUPward 5 degreeUpward 10 degree

(a) Simulated pressure p1

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.5

0

0.5

1

1.5

2

2.5

H=220 mm, p2


(b) Simulated pressure p2

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

H=220 mm, p3


(c) Simulated pressure p3

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

H=220 mm, p4


(d) Simulated pressure p4

Figure 6.23: Simulated pressure for H = 220 mm

81

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

H=110 mm, p1


(a) Simulated pressure p1

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

H=110 mm, p2


(b) Simulated pressure p2

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

H=110 mm, p3


(c) Simulated pressure p3

t√

gH

0 1 2 3 4 5 6 7 8 9 10

PρgH

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

H=110 mm, p4


(d) Simulated pressure p4

Figure 6.24: Simulated pressure for H = 110 mm

6.4 Elevations

6.4.1 Wave probes

Each case is repeated ten times. For instance, Fig. 6.25 shows the experimental

elevation time history at wave probe location h7 and h8 in 10 times repetition for

horizontal bed case H = 110 mm.

time:s0 1 2 3 4 5 6 7 8 9 10

Wate

r le

vel: m

m

20

30

40

50

60

70

80

90

100

110

120wave probe h7 & h8:10 cases

Figure 6.25: Elevation time history of wave probe h7 and h8 for horizontal bed caseH = 110 mm, in 10 repetitions

82

Considering the repetition on the wave probes is quite good, in the followings,

only one typical signal is given to compare with the numerical results.

Fig. 6.26 shows the elevation time history at wave probe location h7 and h8

for the horizontal bed cases. Before the reflection wave advancing back to the wave

probe, a very well agreement can be observed between the OpenFOAM simulation

results (in color blue) and the experimental data (in color red). The difference

between the simulation results and the experimental data mainly lies after the re-

flection wave reaching the wave probe. Even so, the agreement seems to be relatively

well. The differences in h7 and h8 indicates the 3D effects. It is observed that the

3D effects in the numerical simulation results is less than in the experimental data.

time: t√

g/H0 5 10 15 20 25 30 35 40 45

Waterlevel:

h/H

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1H = 110 mm

h7 Experimenth8 Experimenth7 OpenFoamh8 OpenFoam

Refection Wave

(a) H=110 mm

time: t√

g/H0 5 10 15

Waterlevel:

h/H

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1H = 220 mm

h7 Experiment, H=220 mmh8 Experiment, H=220 mmh7 OpenFoamh8 OpenFoam

Refection Wave

(b) H=220 mm

Figure 6.26: Elevation time history at wave probe location h7 and h8 for horizontalbed

The elevation time history at wave probe location h7 and h8 for the slope bed

cases are shown in Fig. 6.27

83

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Downward 10

◦, H = 110 mm


Refection Wave

3D effect

(a) Downward 10◦, H = 110 mm

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1Downward 10

◦, H = 220 mm


Refection Wave

(b) Downward 10◦, H = 220 mm

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1Downward 5

◦, H = 110 mm


Refection Wave

(c) Downward 5◦, H = 110 mm

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Downward 5

◦, H = 220 mm


Refection Wave

(d) Downward 5◦, H = 220 mm

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1Upward 10◦, H = 110 mm


Refection Wave

(e) Upward 10◦, H = 110 mm

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Upward 10◦, H = 220 mm


Refection Wave

(f) Upward 10◦, H = 220 mm

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Upward 5◦, H = 110 mm


Refection Wave

(g) Upward 5◦, H = 110 mm

time: t√

g/H0 2 4 6 8 10 12 14 16 18 20

Waterlevel:h/H

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1Upward 5◦, H = 220 mm


Refection Wave

(h) Upward 5◦, H = 220 mm

Figure 6.27: Elevation time history at wave probe location h7 and h8

84

6.4.2 Virtual wave probes

The image processing for water level detection (WLD) of the virtual wave probes is

described in Appendix C.1. Fig. 6.28 shows the comparison of virtual wave probe h5

between the results of WLD and the previously published experimental data from

LEE et al. [70], BUCHNER [6] and LOBOVSKY et al. [32].

time: t√

g/H0 0.5 1 1.5 2 2.5 3 3.5 4

Wate

r le

vel: h

/H

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35H5

Image Water Level DectectionLobovsky 2014Lee 2002Buchner 2002

Areated area

Figure 6.28: Elevation of virtual wave probe h5

Generally, the results of WLD agree with the other experimental data. However,

when the wave front pass the virtual wave probe, the water level is over-estimated

and with a higher variation. Physically, it is related to the aerated area at the wave

front as shown in Fig. 6.29.

Figure 6.29: Water Level Dectection of h5 (the blue one) at time 120 ms (left) and132 ms (right)

Similarly, Fig. 6.30 shows the WLD results of virtual wave probe h3 and the

corresponded areated peak point at time 224 ms; Fig. 6.31 shows the WLD results

of virtual wave probe h1 with the corresponded small areated front at time 212 ms.

85

time: t√

g/H0 0.5 1 1.5 2 2.5 3 3.5 4

Wate

r le

vel: h

/H

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35H3


Areated area

Figure 6.30: Elevation of virtual wave probe h3 (left) and the corresponded areatedpeak at time 224 ms (right)

time: t√

g/H0 0.5 1 1.5 2 2.5 3 3.5

Wate

r le

vel: h

/H

0

0.05

0.1

0.15

0.2

0.25H1


Areated area

Figure 6.31: Elevation of virtual wave probe h1 (left) and the corresponded smallareated front at time 212 ms (right)

86

Chapter 7

The BIV technique and the

Application

The BIV technique can be used to obtain the velocity field of the dispersed phase.

The technique correlates the fluid particles images and “texture” in the images

created by the air-water interfaces. In this chapter, a test experiment was done.

This Chapter is structured as follows. The shape regimes of fluid particles to-

gether with the dimensionless groups are given first. After that the balance of force

are evaluated with the equation of particle motion, namely the Basset-Boussinesq-

Oseen (BBO) equation. With the BBO equation, the effects of the density ratio

(particle to surrounding fluid) are evaluated. The velocity response for different

particles in different fluid are given. Finally, the main capacity of the BIV system

is formulated.

7.1 BIV test experiment

Figure 7.1: Schematic of the BIV test experiment

The BIV test experiment was constructed and installed in the LabOceano, as shown

in Fig. 7.1. It consists of an acrylic tank and a water pump. The tank is made of

15 mm thick acrylic with inner dimensions of 780×335×600 mm and was mounted

87

on a fixed platform. The water pump with water pipe (diameter of 2.8 cm) were

used to generate a water jet impinging on the water surface. The impinging water

jet entrained the air to the flow and generated lots of bubbles. The illumination

of the flow used the shadowgraph concept. A 500 W (intensity adjustable) halogen

light with a translucent light diffuser were set to the back side of the tank. A high

speed camera was used in the front side to capture the shadow images of the flow.

The distance (L) between the camera and tank was 287 mm. The focal length of

the camera lens was 24 mm and the aperture number was 2.8. The resolution of the

image was 1280× 1024 pix and the frame rate was set to 250 FPS.

Fig. 7.2a shows an example image of bubbles generated by the impinging water

jet. Two plastic rulers were sticked on the tank front surface. As can be observed,

the field of view of image is about 171.3× 136.6 mm. This provides a reference on

the bubble size. To identify the bubbles, a software Image J (SCHNEIDER et al.

[78]) was used. The image processing mainly includes three steps. In Fig. 7.2b, a

FFT band-pass filter was applied to increase the intensity contrast. To remove the

noise part (the rulers) for further processing, the images was cropped. By adjusting

the threshold, the shadow of the bubbles are clearly shown in Fig. 7.2c. Finally,

the bubbles are identified and shown in Fig. 7.2d. The bubble size statistics and

its distribution diagram can then be obtained, as shown in Fig. 7.2e and Fig. 7.2f

respectively. Assuming the bubble size follows the normal distribution, the mean

bubble diameter is about 1.7 mm and the standard deviation is 1.01.

88

(a) Origin image (b) Applied band-pass filter

(c) Cropped and adjusted threshold (d) Bubbles identified

Bubble number ID

0 50 100 150 200 250 300

Bubble

dia

mete

r

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Bubble size distribution

(e) Bubble size statistics

Bubble diameter: mm0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

density

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Bubble size distribution

HistogramNormal

(f) Bubble size distribution diagram

Figure 7.2: An example image and bubble identify process

The main information of the BIV test experiment is summarized in Table 7.1. To

obtain the displacement of the particles, a cross-correlation algorithm is required.

In the present work, we use a GUI-based open-source tool called PIVlab, avail-

able from THIELICKE and STAMHUIS [38]. PIVlab is programmed in MATLAB

and additionally requires the image processing toolbox to run. A PIV analysis

typically consists of three main steps image pre-processing, image evaluation, post-

processing. The overview of the workflow and the implemented features of PIVlab

are summarized in Fig. 7.3. The details of these implemented features are refered to

THIELICKE and STAMHUIS [38] and would not be described here. The quality of

89

the PIV measurements in PIVlab was evaluated using synthetic particle images with

known properties. Under optimal conditions (particle image diameter ≈ 3 pixels,

particle density ≈ 5 - 15 particles/window, no noise, no particle pair loss, no motion

blur, no shear), the bias error of the algorithm is smaller than 0.005 pixels and the

random error is below 0.02 pixels.

Table 7.1: Summary of BIV test experiment

Calibration

Distance of reference points lr 1125 pix

Distance of reference image Lr 150 mm

Magnification factor α 0.134 mm/pix

Field of view 171.3×131.1 mm

Flow Visualization

Average diameter of bubbles 1.67 mm

Camera resolution 1280×1024 pix

Gray scale resolution 8 bit

Frame rate per second 250

Time interval (∆t) 4 ms

Distance of lens to the target lt 287 mm

Length of focus 24 mm

f# number of lens 2.8

Data Processing

Cross correlation method FFT

First pass integration area size 64×64 pix

Second pass integration area size 64×64 pix

Subpixel analysis 3 points Gaussian fitting

Figure 7.3: Workflow of PIVlab, adapted from THIELICKE and STAMHUIS [38]

90

Using the instant of Fig. 7.2a (t0) as reference, the velocity fields in 4 ms obtained

with PIVlab are shown in Fig. 7.4a to Fig. 7.4d, where the time increment equals

to 1 ms. In general, at the right part of these figures, the bubble motion is mainly

downward due to the impinging jet; while at the left part, the bubbles are mainly

rising.

(a) Velocity fields at instant of t0 (Fig.7.2a)

(b) Velocity fields at instant of t0 + 1 ms

(c) Velocity fields at instant of t0 + 2 ms (d) Velocity fields at instant of t0 + 3 ms

Figure 7.4: An example of the output velocity fields

7.1.1 System capacity

The usefulness of a PIV instrument is often characterized by its uncertainty. Since

BIV is a direct adoption of the PIV technique, a brief uncertainty analysis of BIV

is included in Appendix D.1, following the ITTC Recommended Procedures and

Guidelines on PIV uncertainty analysis (PARK et al. [20]). However, as demon-

strated by ADRIAN [79], the uncertainty could not completely describe the capa-

bilities of the system. There are another two related and equally important charac-

teristics: dynamic velocity range (DVR) and dynamic spatial range(DSR).

The DVR is defined as the ratio of maximum velocity (umax) that can be mea-

91

sured to minimum resolvable velocity.

DV R =umaxσu

(7.1)

where σu is the minimum resolvable velocity, can be evaluated by the uncertainty

analysis on u. It should be noted that the relative error in velocity can be evaluated

as the reciprocal of DVR, e.g. for a system with DVR larger than 20, the relative

error on u is less than 5%.

The maximum velocity can be calculated as,

umax =∆xmax

∆t=

∆Xmax

M∆t(7.2)

wher ∆x and ∆X are the displacements, lower case letters refer to quantities in

the fluid and upper case letters denote quantities on the image and M is the mag-

nification factor, in pix/mm. To avoid loss of correlation due to excessive in-plane

displacement, KEANE and ADRIAN [80] state that the displacement should be

smaller than one quarter of the interrogation window size DI :

∆Xmax

DI

≤ 1

4(7.3)

Thus the maximum displacement could be assumed as ∆Xmax = 14DI . In our setup,

the images are captured by the high speed camera. The time interval between two

neighbor frames can be assumed as uniform, thus ∆t can be evaluated with the

reciprocal of frames per second (FPS). It then gives,

DV R =FPS DI

4Mσu(7.4)

The DSR can be defined as the field-of-view (FOV) in the object space divided by

the smallest resolvable spatial variation.

DSR =lx

∆xmax(7.5)

where lx is FOV in x direction and minimum resolvable spatial variation can be

estimated as ∆xmax. If the working distance of the camera is constrained, lx is

proportional to the camera sensor chip size Sx. To have a large DVR on a BIV

system, it is recommended to use a camera with large chip size.

For the BIV test experiment, Fig. 7.5a shows the relationship between the uncer-

tainty on u with the average bubble diameter dp and Fig. 7.5b shows the relationship

of the dynamic range (DVR and DSR) against the average bubble diameter. For

dp = 1.7, the uncertainty on velocity (σu) is about 60.64 mm/s, DVR is about 13,

92

DSR is about 53. Generally, the uncertainty and DVR is a function of average bub-

ble diameter. From the uncertainty analysis, it can be found that in BIV the bubble

diameter is much larger than the seeding particles used in PIV and it becomes a main

error source. Generally, to improve the uncertainty, the best way is using the BIV

technique for small bubbles. In addition, small bubble diameter means large DVR

while DSR is independent of bubble diameter and related to camera characteristics

(the sensor size).

bubble diameter: dp (mm)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

uncert

ain

ty: σ

u (

mm

/s)

0

10

20

30

40

50

60

70

80

(a) Uncertainty and average bubble di-ameter

bubble diameter: dp (mm)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

20

30

40

50

60

70

80

90

100

DVRDSR

(b) Dynamic range and average bubblediameter

Figure 7.5: Results for the test experiment

7.2 Considerations on bubble/drops properties

In contrast to the PIV technique using the solid seeding particles, the concept of the

so-called BIV technique proposed by RYU et al. [15] used the bubble itself as tracer.

The characteristics of the bubble are quite different from the solid particles. This

requires a good understanding on the properties on the dispersed phase. Following

common usage, dispersed phase is a distinct state of matter (the particles) in a

system, separated from other material by the phase boundary and the ”continuous

phase” is refer to the fluid surrounding the particles.

In this work, the ”particle” is some kind self-contained body, with dimension

range about 0.5 µm (PIV seeding particles) to several millimeters (large air bub-

bles or drops), separated from the surrounding fluid by a recognizable interface.

”solid particles”, ”bubbles”, ”drops” are particles whose dispersed phases are in the

solid state, gas state and liquid state respectively. Together, ”bubbles” and ”drops”

comprise the ”fluid particles”. As documented in RAFFEL et al. [16], the main re-

quirements on PIV seeding particles are related to the following properties, namely

particle dimension size, particle density, particle light scatter characteristic, particle

distribution. As discussed in the previous section, the specific light scatter charac-

teristic of bubble has been taking into consideration by the shadowgraphy technique.

93

Following the concept of PIV, the particle density distribution are assumed to be

medium. Beside this, the particle dimension size and particle density are the main

aspects with regards to measurement accuracy and would be explored in the present

Chapter. To distinguish the properties of the dispersed (or particle) phase from the

continuous (or fluid) phase, subscripts ”p” and ”f” are used respectively.

7.2.1 Shape regimes of the fluid particles

In the absence of other forces or constraints, the action of surface tension on bubbles

and drops tends to preserve their spherical shape. However, under the influence

of gravitational forces, bubbles and drops in free rise or fall in infinite media are

generally grouped in three different regimes CLIFT et al. [81]: spherical, ellipsoidal,

and spherical cap, as shown in Fig. 7.6.

Figure 7.6: Shape regimes for bubbles and drops, adapted from CLIFT et al. [81]

Following AMAYA-BOWER and LEE [82], a brief description of the three main

regimes provided as follows:

• Spherical regime:

This regime is dominated by surface tension and viscous forces. Original size of

the bubble is small, usually less than 1.3 mm. At low ReT (ReT < 1), bubbles

and drops are spherical , regardless of the value of Bo; while at intermediate

values of Ret, bubbles and drops are spherical only when Bo < 1.

94

• Ellipsoidal regime:

This regime is mainly dominated by surface tension. Bubble size is intermedi-

ate, typically from 1.3 to 6 mm. The range of Bond number is 0.25 < Bo < 40

and the ReT is usually large than 1.

• Spherical cap regime:

This regime is governed by inertia force. Bubble size is large, usually bigger

than 6 mm, with relative high Bond number Bo > 40 and Reynolds number

ReT > 1.2.

These regimes are depended on the following dimensionless groups, the Bond

number (Bo) 1, the Morton number (Mo) and the terminal Reynolds number (ReT )

given by:

Bo =g∆ρd2

p

σ(7.6)

Mo =gµ4

f∆ρ

ρ2fσ

3(7.7)

ReT =ρfdpUTµf

(7.8)

where ρf is the fluid density, ∆ρ = ρf − ρp is the density difference between the

continuous phase (fluid) and the dispersed phase (particles), µf is the fluid viscosity,

σ is the surface tension, g is the gravitational acceleration, and UT is the terminal

velocity of the fluid particles, defined as the steady velocity that the bubbles or

drops reaches when there is a balance between buoyancy and drag forces.

The Bond number is the ratio of body forces and surface tension foces. Since

Bond number is a funtion of the bubble diameter, proportional to d2p, it might also

be considered as a dimensionless value of the particle size. The relationship between

the fluid particles(bubbles/drops) diameter and the Bond number is shown in Fig.

7.7. Morton number provides a description of the properties of the surrounding fluid,

mainly focusing in viscosity and surface tension. If the fluid properties are given,

Morton number is a constant not affected by the particle diameter. The terminal

Reynolds number depending on the terminal velocity is given the following section.

Terminal Reynolds number

According to CROWE et al. [83], the terminal velocity, UT , is the final velocity a

particle attains falling in a quiescent fluid. Various formulations of the terminal

velocity have been presented in the literature.

1The Bond number is sometimes reffered to as Eotvos number.

95

Bubbles/Drops diameter (mm)0 1 2 3 4 5 6 7 8 9 10

Bo

10-3

10-2

10-1

100

101

102

Bond Number

Air bubble in waterWater drop in air

Figure 7.7: Bond number versus bubbles or drops diameter

One approach could follow AMAYA-BOWER and LEE [82], the terminal velocity

is a combination of three main different shape regimes repectively to CLIFT et al.

[81],

• Spherical regime (typically dp < 1.3mm)

The terminal velocity is proportional to size of the bubble, which can be de-

scribed by HADAMARD [84] and RYBCZYNSKI [85] as follows:

UT =g∆ρd2

p

6µ

1 + κ

2 + 3κ(7.9)

where κ =µfµg

is density ratio.

• Ellipsoidal regime:

Terminal velocity is closely approximated by a correlation suggested by

MENDELSON [86] based on wave theory, given as:

UT =

√2.14σ

ρdp+ 0.505gdp (7.10)

• Spherical cap regime:

Terminal velocity of this regime can be described by a theory proposed by

DAVIES and TAYLOR [87]:

UT =2

3

√g∆ρdp

2ρ(7.11)

96

Another approach is an empirical relationship that fits the data for all three

main regimes, as described in CROWE et al. [83]:

ReT =

(√

22 +√

4.89Ga−√

22)2 for Ga < 4× 105

1.74√Ga for 4× 105 < Ga < 3× 1010

(7.12)

where Ga is the Galileo number, defined as,

Ga =gd3

p(ρpρf− 1)

ν2f

(7.13)

The terminal Reynolds number obtained from both approaches are shown in Fig.

7.8. To void the discontinuity, in the present work, the formulations by CROWE

et al. [83] are used.

Bubbles/Drops diameter (mm)0 1 2 3 4 5 6 7 8 9 10

Re

T

10-1

100

101

102

103

104

105

Terminal Reynolds Number

Air bubble in water,Amaya-Bower and Lee 2010Air bubble in water,Crowe et,al 2011Water drop in air,Amaya-Bower and Lee 2010Water drop in air,Crowe et,al 2011

Figure 7.8: Terminal Reynolds number versus bubbles or drops diameter

In the present work, we considered the fluid particle (bubble/drops) dimension is

up to several millimeters. From Fig. 7.6, the shape regimes are mainly spherical and

ellipsoidal. Assuming the particle diameter of a ellipsoidal type can be estimated as

a sphere with equivalent volume.The smaller the particle diameter, the closer to a

sphere.

7.2.2 Equation of particle motion

One very important consideration in the PIV technique is that the seeding particles

can follow the flow motion. However, the bubble motion is very sensitive to excita-

tion by external forces at high frequencies owing to the small inertia compared to

97

the surrounding liquid. Questions may arise on whether the bubbles follow the fluid

or not. In this section, we explore the equation of particle motion to evaluate the

particle response in fluid.

Within the approximation of low Reynolds number, the equation of motion of a

small spherical particle immersed in a fluid flow is given by MCKEON et al. [88] as:

ρpπd3

p

6

dvpdt

=πd3

p

6(ρp − ρf )g︸︷︷︸term 1

− 3πµdp[(vp − U)− 1

24d2p∇2U ]︸︷︷︸

term 2

+πd3

p

6ρf

dU

dt︸︷︷︸term 3

− π

12ρfd

3p

d

dt[(vp − U)− 1

40d2p∇2U ]︸︷︷︸

term 4

− 3

2d2p

√πµρf

∫ t

t0

d

dξ[(vp − U)− 1

24d2p∇2U)]

dξ√t− ξ︸︷︷︸

term 5

(7.14)

Eq. (7.14) is known as the Basset-Boussinesq-Oseen (BBO) equation. It describes

the motion of - and forces on - a small particle in unsteady flow at low Reynolds

numbers. The equation is named after Joseph Valentin Boussinesq, Alfred Barnard

Basset and Carl Wilhelm Oseen. The first term in Eq. (7.14) is the body force

(gravity minus buoyancy). The second term is a quasi-steady viscous force. The

third term is the Froude-Krylov force due to the pressure gradient in the undisturbed

flow. The fourth term is the added-mass force which is purely inertial. The fifth

term is called the history force or Basset force. It is due to the diffusion of the

vorticity from the particle surface to the bulk fluid flow.

The details of the derivation of the BBO equation can be found in CROWE et al.

[83] and the solution of the BBO equation has been given by HJELMFELT JR

and MOCKROS [89]. A more recent discussion of this equation and solution for

turbulent flow has been given by MEI [90].

The ∇2U terms in Eq. (7.14) are normally small in comparison with any of the

remaining terms and can be neglected. For very small particle tracers as used in

PIV, the first part of the quasi-steady viscous force, the Stokes drag, dominates the

right-hand side of the equation. The difference between the particle velocity vp and

98

that of the surrounding fluid U can be estimated as,

vp − U =d2p(ρp − ρf )

18µ

dvpdt

(7.15)

Clearly from the choice of neutrally buoyant particles ρp ≈ ρf leads to particle

tracers that accurately follow the flow that vp ≈ U .

To simplify the problem, the unsteady forces (the Basset history force), the

higher order term (∇2U) and the body force term are ignored, Eq. (7.14) becomes,

ρpπd3

p

6

dvpdt

=πd3

p

6ρf

dU

dt− 3πµdp(vp − U)− π

12ρfd

3p

d(vp − U)

dt(7.16)

To examine steady-state particle response to the oscillating flow of arbitrary sum

of frequencies, we assume that velocity U can be represented as an infinite sum of

harmonic functions as:

U(t) =

∫ ∞0

Λf (w) exp(iwt)dw

dU(t)

dt=

∫ ∞0

iwΛf (w) exp(iwt)dw

vp(t) =

∫ ∞0

Λp(w) exp(iwt)dw

dvp(t)

dt=

∫ ∞0

iwΛp(w) exp(iwt)dw

(7.17)

Substituting Eq. (7.17) in Eq. (7.16), yields∫ ∞0

{StΛpi− StγΛf i+ (Λp − Λf ) +St2γ(Λp − Λf )i} exp(iwt)dw = 0 (7.18)

where St is the Stokes number defined as the ratio of the characteristic time of a

particle (or droplet) τp to a characteristic time of the fluid τf and γ is the the density

ratio,

St =τpτf

=ρpd

2pw

18µ(7.19)

γ =ρpρf

(7.20)

The Stokes number is a very important parameter in fluid-particle flows. If St � 1,

the response time of the particles is much less than the characteristic time associated

with the flow field. The particles will have enough time to respond to changes in

flow velocity. Thus the velocity of the particle and the fluid velocity will be nearly

equal. On the other hand, if St � 1 then the particle will essentially have no time

to respond to the fluid velocity changes. Thus particle velocity will be affected and

99

not equal to the fluid velocity.

By solving Eq. (7.18) yields,

Λp

Λf

=1 + i3Stγ

2

1 + i(St + Stγ2

)(7.21)

the velocity amplitude response could be defined as,

|vpU| =

√ΛpΛ∗pΛfΛ∗f

(7.22)

Stokes Number10

-510

-410

-310

-210

-110

010

110

210

310

410

5

AmplitudeResponse

=|u

p/u

f|

10-3

10-2

10-1

100

101

Air bubble in waterHollow glass spheres (1100kg/m

3) in water

TiO2 (3950kg/m3) in water

Water drop in air

Figure 7.9: Velocity amplitude response versus Stokes number

Fig. 7.9 shows the relationship between the velocity amplitude response and the

Stokes number. The velocity amplitude response could be used to quantify whether

a particle follows the fluid or not. As one can imagine, light particles will follow

the fluid better than the heavier particles. For common seeding particles applied

in the PIV technique, the density ratio is usually larger than one (γ < 1). As can

be observed in Fig. 7.9, the velocity response |vpU| < 1 and this is usually called

“velocity lag”. Comparing two type of seeding particles, the Hollow glass spheres

(1100 kg/m3) and the Titanium Dioxide (TiO2) spheres (3950 kg/m3), the velocity

amplitude response of the light particles (Hollow glass spheres) is closer to one. The

closer of the particle density to the fluid density, the better in following the flow.

When the particle density is not close to the fluid density, whether the particles

100

can be considered as following the flow is regards to the Stokes number. For heavier

particles (γ � 1), e.g. water drops in air, if the Stokes number is less than 0.1, it may

be regards as following the flow. On the other hand, for very light particles(γ � 1),

e.g. air bubbles in water, the situation is very different. The velocity response is

generally larger than one (|vpU| > 1), which indicates that the bubble’s velocity is

generally over-response of the fluid. Only when the Stokes number is very small,

less than 10−4, the bubble can be regards as a good tracer.

7.3 Application of the BIV technique

Taking the advantage of the BIV technique, it is possible to obtain more information

on the flow velocity field. In this section, we show some example results obtained

using the BIV technique.

101

7.3.1 Example in the main period

(a) (b)

(c) (d)

(e) (f)

Figure 7.10: H=110 mm, BIV analysis velocity vectors in time 12, 20, 28, 36, 44, 52ms

In the main period, the velocity field during the gate releasing stage can be obtained

using the BIV technique. The velocity vectors of case H = 110 mm are shown in

Fig. 7.10. It can be observed that the BIV technique detects not only on the flow

velocity field but also on the gate releasing movement. The gate releasing velocity

in time history is then shown in Fig. 7.11. It shows that once the gate was released,

the gate velocity start to increase rapidly and later due to the non-uniform friction

between the gate and the tank, there is a small variation of gate velocity. If the first

4 points in Fig. 7.11 are not accounted, the mean velocity equals to 2.11 m/s which

is close to the average velocity Vg in Table 6.1 of case H = 110 mm.

102

t: ms

0 5 10 15 20 25 30 35 40 45 50

V:m/s

0

0.5

1

1.5

2

2.5

BIV gate velocityMean velocity except the first 4 points

Figure 7.11: Gate releasing velocity in time history obtained with BIV

The front velocity is an important characteristic of the dam break wave. Since

the flow is advancing in channel, the wave front is also developing in horizontal

direction. Most velocity measurement techniques require to be mounted on certain

location to measure the flow velocity on a single point, such as Pitot-tube, Hot Wire

Anemometry (HWA) and Laser-Doppler Anemometry (LDA). They can not track

the advancing wave front. Now with the BIV technique, it is possible to track the

wave front velocity in time evolution, as shown in Fig. 7.12.

(a) At time t = 158 ms and t∗ = 1.50 (b) At time t = 198 ms and t∗ = 1.89

Figure 7.12: BIV results example during the runup

The results obtained by BIV technique are shown in Fig. 7.13, compared with

numerical simulation results and analytical formulations summarized in Table 7.2.

103

t√

g/H0 0.5 1 1.5 2 2.5

U/√

gH

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

BIV H=220 mmBIV H=110 mmOpenFoam H=220 mmOpenFoam H=110 mmDressler 1954, f=0.04Whitham 1955, f=0.04Chanson 2005, f=0.04t1

Figure 7.13: Wave front velocity time evolution

Table 7.2: Analytical approaches of front velocity with bottom friction effects

Reference Asymptotic solution Remarks

DRESSLER [31] UF√gH

= 2− 3.59(f8

√gH

)13 first order approximation

WHITHAM [18] UF√gH

= 2− 3.452(f8t√

gH

)13 first order approximation

CHANSON [19] 13

(1− UF2√gH

)3

f8

U2F

gH

= t√

gH

exact solution

It can be observed that although there is some variation in the the front velocity

results extracted with BIV technique, the general trends of the BIV results agree

with OpenFOAM simulation results for both cases of H = 110 mm and H = 220

mm. Initially the water is at rest, once the dam gate is released, the front velocity

starts increasing. The main acceleration range may be considered as interval [0, t∗1],

as can be observed form Fig. 7.13. During this interval, the wave front velocity

increases rapidly and after time t∗1, the front velocity increases slowly. Taking the

advantage of BIV technique, the average dimensionless velocity of the downstream

wave front after time t > t∗1 are obtained. The results are shown in Table 7.3, in

comparison to previously published data.

104

Table 7.3: The average dimensionless wave front velocity for time t > t∗1

Reference and case (mm) U/√gH

Present case H=110 1.33

Present case H=220 1.41

LOBOVSKY et al. [32] H=300 1.56

LOBOVSKY et al. [32] H=600 1.34

MARTIN and MOYCE [91] H=57 1.48

MARTIN and MOYCE [91] H=114 1.69

In the classical ESS, the bottom friction is neglected and the downstream wave

front velocity is constant equals to 2√gH. However, in reality, due to the bottom

friction effects, the wave front velocity is not constant but a function of time, as can

be observed from Fig. 7.13. These analytical formulations in Table 7.2 include the

bottom friction effects (represented by f , as described in Section 3.2.1), but they

are all based on the ESS. Thus it is not surprised that they all have the common

discontinuity at time t = 0+, the wave front velocity jumps from zero to 2√gH. Due

to this discontinuity, it is could be observed that these formulations overestimate

the wave front velocity in the interval [0, t∗1] too much; while for time larger than t∗1,

they are closer to BIV results and simulation results.

7.3.2 Example in the impact period

In the impact period, the BIV technique may be used to evaluate the runup process.

Fig. 7.14 shows the example results at instant 358 ms and 382 ms obtained by the

BIV technique. The main runup velocity field are captured, including the solid flow

(the continuious phase) and some splashing water drops (the dispersed phase). It is

should be noted in the left part of the tank, there are many drops the velocity field

of which not captured. These water drops are mainly attached on the inner tank

surface. The velocity of these water drops is relatively small to the uncertainty of

BIV system and thus can not be captured by the algorithm.

105

(a) At time t = 358 ms and t∗ = 3.41 (b) At time t = 382 ms and t∗ = 3.64

Figure 7.14: BIV results example during the runup

7.3.3 Example in the reflection period

After the runup, the fluid is falling down. At time 646 ms, a breaking wave is

generated and a vortex is detected, as shown in Fig. 7.15a. Due to the breaking

wave advancing back, large mixed air-water area is generated. As shown in Fig.

7.15b, at time 752 ms, BIV technique detects velocity field including: falling water

drops, bubbles, mixed air-water area and free surface.

(a) At time t = 646 ms (b) At time t = 752 ms

Figure 7.15: BIV results example of the reflection wave

7.3.4 Notes on BIV technique application

It is should be noted that the original application of PIV/BIV is for the velocity

fields for particles. However, it is found that the technique could be extended for

falling water drops, bubbles, mixed air-water area and free surface as shown in Fig.

7.15b or even for dam gate releasing as shown in Fig. 7.10.

The reason of this extention may be explained as follows. According to HUANG

et al. [92] and HUANG et al. [37], the cross-correlation algorithm of PIV is actually

a image pattern matching technique. The image pattern is an image (of particles)

106

with a certain spatial distribution in a given space. This indicates that the cross-

correlation algorithm is not only limited for small solid seeding particles but may

also suitable for any certain distribution of objects. The object could be a group of

deformable bubbles (or drops) and in a very short time (the time interval between a

image pair) we assumes the deformation is neglectable, this yilds the BIV technique.

The object could also be the shadow image of the mixed air-water area, the shadow

image of a thin free surface or even the image of a moving gate.

107

Chapter 8

Conclusions and Future Work

8.1 Conclusions

The green water generated by extreme waves is primarily an air-water mixture, which

includes a continuous phase and a dispersed phase. In this thesis, the continuous

phase is mainly studied with the dam break model and the dispersed phase is mainly

studied with the BIV technique.

8.1.1 Continuous phase

In the continuous phase, to introduce the bottom friction and the bed slope effects

for shipping of water problem, the dam break model is explored by three main

approaches, namely analytical, numerical and experimental.

• Analytically, an extended analytical dam break solution is proposed. The

presented solution is a piecewise solution (PS) and consists of three parts:

upstream part, downstream frictionless part and downstream wave tip part.

PS assumes that incoming wave always comes horizontally and induces a hor-

izontal dam in the upstream side. In the downstream side, PS assumes that

the downstream elevation of a sloping bed case can be transformed from the

elevation of the horizontal bed case.

• Numerically, the dam break simulations are based on the open source tool

OpenFOAM version 2.3.0. The numerical dam break model solves the RANS

equations with a k − ε turbulence model.

• Experimentally, different dam break experiments were carried out, with one

horizontal bed and four slope beds with different slope angles. Each bed was

tested with two different water levels.

108

The acquired experimental data and the simulated numerical results agree well

with the presented piecewise solution. Besides, based on the research presented in

this thesis, the following conclusions seem to be justified:

• The initial stages of dam break is studied. The dam break period (before

impact) could be divided into three different stages: the gate releasing stage,

the transition stage and the later stage. The gate releasing stage is dominated

by the vertical motion and this stage is better described with the Lagrangian

Stoker solution; while the later stage of dam break wave is dominated by

horizontal motion and this stage is better represented with the Eulerian Stoker

solution. During the gate releasing stage, the “sudden dam break” concept is

revised. The dam may be considered break suddenly when the dimensionless

gate removal period is smaller than3t∗15≈ 0.63.

• By introducing a time compensation factor t∗c = t∗1/3 into the PS, the tracking

ability of downstream dam break wave front is greatly improved, where t1

is the instant when the Eulerian and Lagrangian Stoker solutions have an

intersection point at the dam section. The predicted downstream wave front

by the PS agree well with experimental data on both wave front convex shape

and the front location. Since t∗c is introduced in the solution, the solution

would not work for t∗ < t∗c . We recommend to use PS with tc for time large

than3t∗15

, that is after the “sudden dam break”.

• To avoid the turbulent wakes induced by the wave probes in the downstream

side, we introduced the virtual wave probes with a Water Level Detection

(WLD) algorithm. A good agreement is found between the results of virtual

wave probes and previously published data.

8.1.2 Dispersed phase

In the dispersed phase, the main goal is to evaluate the BIV technique. The special

bubbles/drops characteristics about the BIV technique are evaluated. Mainly two

characteristics of bubbles/drops are discussed: the particle density and the particle

diameter. Based on the research presented in this thesis, the following conclusions

seem to be justified:

• The density ratio (γ) between the particle and fluid affects the particle ve-

locity response in fluid. The velocity response of bubbles/drops in the fluid

is evaluated with the BBO equation. As shown in Fig. 7.9, in general, the

velocity amplitude response between the particle and fluid is less than one for

heavier particles (γ � 1, e.g. water drops in air), and larger than one for

109

lighter particles (γ � 1, e.g. air bubbles in water). Only when the Stokes

number is sufficient small, the bubbles/drops can be regards as a good tracer

of the fluid.

• The bubbles/drops diameter is another important factor for the BIV technique.

By analyzing the shape regimes of bubbles/drops, it is found particles with

the small diameter are closer to a spherical ideal tracer. Following the ITTC

Recommended Procedures and Guidelines on PIV uncertainty analysis (PARK

et al. [20]), it is found the bubble diameter is a very important factor which

affects the uncertainty. What’s more, the Stokes number is also a function of

the particle diameter.

• The BIV technique is also applied to the dam break experiment. It is found

that the technique could be used for the velocity measurement of falling water

drops, bubbles, mixed air-water area and even gate releasing. For instance, the

BIV technique can be used to measured wave front velocity. A good agreement

is fount between BIV results and the numerical simulation data.

8.2 Future works

The relative motion between the ship and the incoming wave is an interesting but

complicated topic. In the present work, the influence of the pitch motion is evaluated

by a static pitch angle together with the dam break model. Further studies may

consider more on the dynamic behavior of the pitch motion.

In the dam break model, the effects of the bottom friction are taking account

with a constant friction factor. Although there are several empirical formulations to

estimate the bottom friction factor, but these formulations works mainly for steady

flow. For unsteady flow like the dam break wave, the investigation is not clear yet

and requires further studies.

The classical PIV is applicable for the 2D velocity measurement. With the re-

cent developments, the PIV technique now towards to the 3D measurement. There

are various approaches developed to resolve the 3D velocity information. For in-

stance, the defocusing PIV technique developed by PEREIRA et al. [93], PEREIRA

and GHARIB [94], the synthetic aperture particle image velocimetry developed by

BELDEN et al. [95], or the Tomographic PIV developed by ELSINGA et al. [96].

With the advances in the PIV technique, the BIV technique may also be extended

into 3D velocity measurement.

110

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Appendix A

Shallow Water Equations

A.1 Shallow Water Equations

The Shallow Water Equations (SWE) can be derived from equations of conservation

of mass and conservation of momentum,the Navier-Stokes Equations.

∇ · u = 0 (A.1)

∂u

∂t+ u · ∇u = g − ∇p

ρ+ µ∇2u (A.2)

with the following assumption:

• Two-dimensional problem

• Newtonian in-compressible fluid

• Hydro-static pressure

• Body force only gravity

• Depth averaged velocity

• Viscous effects ignored

Conservation of mass

As shown in Fig. A.1,

ξ = ξ(x, y, t) is the elevation of the free surface;

b = b(x, y) is the bed slope,positive downward;

h = h(x, y, t) is the total depth of the water.

Note that h = ξ + b.

120

Figure A.1: A Typical Water Column

Now integrate the continuity Eq. A.1 from z = −b to z = ξ. Since both b and ξ

depend on t, x, and y, we apply the Leibniz integral rule: 1

0 = ∇ · v

=

∫ ξ

−b(∂u

∂x+∂v

∂y)dz + w|z=ξ − w|z=−b

=∂

∂x

∫ ξ

−budz +

∂

∂y

∫ ξ

−bvdz − (u|z=ξ

∂ξ

∂x+ u|z=−b

∂b

∂x)

− (v|z=ξ∂ξ

∂y+ v|z=−b

∂b

∂y) + w|z=ξ − w|z=−b

(A.3)

Defining depth-averaged velocities as:

u =1

h

∫ ξ

−budz, v =

1

h

∫ ξ

−bvdz (A.4)

Applying Boundary Conditions to get rid of the boundary terms.

at the free surface,

w =∂ξ

∂t+ u

∂ξ

∂x+ v

∂ξ

∂y(A.5)

at the bottom,no slip condition,

u = v = w = 0 (A.6)

So the depth-averaged continuity equation is:

∂ξ

∂t+∂hu

∂x+∂hv

∂y= 0 (A.7)

1Let f(x, θ) be a function such that fθ(x, θ) exists, and is continuous.Then,d

dθ

(∫ b(θ)a(θ)

f(x, θ) dx)

=∫ b(θ)a(θ)

∂θf(x, θ) dx+ f(b(θ), θ)b′(θ)− f(a(θ), θ)a′(θ)

121

With the horizontal bottom, Eq. A.7 could be written as,

∂h

∂t+∂hu

∂x+∂hv

∂y= 0 (A.8)

Conservation of momentum

Momentum Eq. A.2 in x direction could be written as,where τvis represents the

viscous terms,∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −1

ρ

∂p

∂x+ τvis (A.9)

Integrating the left-hand side of the x-momentum equation over depth, we get:∫ ξ

−b[∂u

∂t+∂u2

∂x+∂uv

∂y]dz =

∂hu

∂t+∂hu2

∂x+∂huv

∂y+ {dif.adv.terms} (A.10)

The differential advection terms account for the fact that the average of the product

of two functions is not the product of the averages.(uv 6= uv)

Integrating the right-hand side of the x-momentum equation over depth, and

assuming hydrostatic pressure p = ρgh, we get:∫ ξ

−b(−1

ρ

∂p

∂x+ τvis)dz = −gh∂h

∂x+

∫ ξ

−bτvisdz (A.11)

Combining the LHS and RHS of the depth-integrated x-momentum equa-

tion,ignoring the differential advection terms and the viscous terms, we get:

∂hu

∂t+∂hu2

∂x+∂huv

∂y= −gh∂h

∂x(A.12)

The result for y-momentum equation is similar,

∂hv

∂t+∂huv

∂x+∂hv2

∂y= −gh∂h

∂y(A.13)

Combining the depth-integrated continuity Eq. A.8 with the depth-integrated x-

momentum and y-momentum equations, the 2-D (nonlinear) Shallow Water Equa-

tions in non-conservative form are:

∂h

∂t+∂hu

∂x+∂hv

∂y= 0

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −g∂h

∂x∂v

∂t+ u

∂v

∂x+ v

∂v

∂y= −g∂h

∂y

(A.14)

122

A.1.1 Saint Venant Equations

The 1-D form of Shallow Water Equations is also known as Saint Venant Equations

(SVE). Where,the viscous term represents by the friction slope factor Sf , and gravity

body force represents by the bottom slope factor S0 (S0 = sin θ):

∂h

∂t+∂hu

∂x= 0 (A.15)

∂u

∂t+ u

∂u

∂x+ g

∂h

∂x= g(S0 − Sf ) (A.16)

A.1.2 Method of Characteristics

A mathematical technique to solve the SVE is the method of characteristics. By

introducing the propagation speed c =√gh, the SVE could be written as,

2ct + 2ucx + cux = 0

ut + uux + 2ccx = g(S0 − Sf )(A.17)

The subtraction and addition of the partial differential Equations A.17 yields to

ordinary differential equations called the Characteristic Differential Equations,

[∂

∂t+ (u− c) ∂

∂x] · [u− 2c− g(S0 − Sf )t] = 0

[∂

∂t+ (u+ c)

∂

∂x] · [u+ 2c− g(S0 − Sf )t] = 0

(A.18)

Which leads to a characteristic system of two families: characteristic C1 and char-

acteristic C2,

C1 :dx

dt= u+ c

C2 :dx

dt= u− c

(A.19)

along with the relations,

u+ 2c− g(S0 − Sf )t = const. along curve C1

u− 2c− g(S0 − Sf )t = const. along curve C2

(A.20)

The solution can be integrated from this characteristic system with the initial data.

123

Appendix B

Camera Calibration

B.1 camera calibration

In the present work, the camera was calibrated with a checkerboard pattern with

square size equal to 40 mm. A set of calibration images with the checkerboard in

different locations was recorded and one is shown in Fig. B.1a.

The calibration code developed for the present work was based on the OpenCV

Toolbox by BRADSKI and KAEHLER [97]. The code first detects the checkerboard

in the images and then calculates reprojection errors by projecting the checkerboard

points from world coordinates (defined by the checkerboard) into image coordinates.

The detected checkerboard points are represented in green circles while the red cross

indicates the reprojected points, as shown in Fig. B.1b. The camera calibration

accuracy can be examined by the reprojection errors.

(a) original image

Detected PointsReprojectedPoints

(b) reprojected image

Figure B.1: camera calibration image example

The reprojection errors are the distances in pixels between the detected and the

reprojected points. Reprojection errors provide a qualitative measure of accuracy.

As a general rule, reprojection errors of less than one pixel are acceptable. As shown

124

in Fig. B.2, the mean reprojection error here is 0.27 pixel, considered to be small.

Images0 5 10 15 20 25

Mean E

rror

in P

ixels

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Mean Reprojection Error per Image

Overall Mean Error: 0.27 pixels

Figure B.2: camera calibration mean error

The relative positions between the camera and the checkerboard in the used

calibration image sets are shown in Fig. B.3.

X (mm)

100

-100

0

Xc

-200

Yc

Zc

0

500

Extrinsic Parameters Visualization

1000

Z (mm)

1500

2000

2500

1011121314657483291151617181920

3000

2122

3500

400

300

200

100

0

Y (

mm

)

Figure B.3: camera calibration mean error

It is also important to note the calibration algorithm uses the camera model

proposed by BOUGUET [98] which includes the radial and tangential lens distortion.

The radial distortion occurs when light rays bend more near the edges of a lens than

they do at its optical center. The smaller the lens, the greater the radial distortion.

Tangential distortion occurs when the lens and the image plane are not parallel.

125

After calibration, all the images have been corrected for lens distortion. With the

calibration results, the transformation from physical coordinates to the image coor-

dinates was built. This allows further processing, for instance, plotting analytical

solutions or numerical results over the experimental images.

126

Appendix C

Water Level Dectection

C.1 Algorithm for water level detection

The image processing of water level dectection is done with Computer Vision System

Toolbox in Matlab version 2014 b. The algorithm mainly includes the following

steps:

1. Image undistortion

2. Convert image to grayscale level

3. Edge detection

4. Increase the thickness of the dectected edge to enhance the performance of the

Hough transformation

5. Find the longest line with Hough transformation in the edge image

First the image is undistorted based on the camera calibration as shown in

Section B.1 and then converted to the grayscale level if necessary. The As shown

in Fig. C.1, due to using the ”AF DC-NIKKOR 105mm f/2D“ lens, the image

undistortion is quite small in this project. It can also be observed that in the

downstream side, there are three black thin markers, which are called virtual wave

probe in Section 5.3.1.

To dectect the water level at the given location, where the virtual wave probes

located, the key is to identify the water level between the edge of the free surface and

virtual wave probe. The edge dectection method is therefore applied. According

to JAHNE [99], mathematically speaking, an ideal edge is a discontinuity of the

spatial gray value function g(x) of the image plane. Edges characterize boundaries

and are a problem of fundamental importance in image processing and particularly

in automatic feature extraction. There are various edge detection algorithms. The

127

(a) Original Image (b) Undistorted Image

Figure C.1: Original

comparison between commonly used edge detection algorithms such as Sobel, Canny,

Prewitt, Roberts, Laplacian and Zero Crossing could be found in KATIYAR and

ARUN [100]. In general, Canny method performance better but its computational

complexity is higher. The Sobel method also detects various features as Canny

method but computationally more efficient. The other algorithms as Robert and

Prewitt also detect the various features but fails in case of smaller features and the

range of usable threshold is very low. Therefore, the Sobel method is choosed as the

edge detection algorithm.

Figure C.2: Edge dectected with Sobel method

The results of Soble method is shown in Fig. C.2, the main edges features have

been captured. It is also clealy observed that the virtual wave probes lines break

at the free surface. It is possible to measure the water level based on the wave

probe line length. To identify those virtual wave probe lines, the Standard Houth

transform (SHT) is applied. To improve the performance of SHT, the egde thickness

128

is dilated before the SHT algorithm, as shown in Fig. C.3

Figure C.3: Edge thickness dilated

The SHT uses the parametric representation of a line:

ρ = x cos(θ) + y sin(θ) (C.1)

The variable ρ is the distance from the origin to the line along a vector perpendicular

to the line. θ is the angle of the perpendicular projection from the origin to the

line measured in degrees clockwise from the positive x-axis. The range of theta is

90◦ ≤ θ < 90◦. The angle of the line itself is θ + 90◦, also measured clockwise with

respect to the positive x-axis.

Line

Figure C.4: Hough transform for straight lines

The SHT is a parameter space matrix whose rows and columns correspond to

ρ and θ values respectively. The elements in the SHT represent accumulator cells.

Initially, the value in each cell is zero. Then, for every non-background point in the

image, ρ is calculated for every θ. ρ is rounded off to the nearest allowed row in

SHT. That accumulator cell is incremented. At the end of this procedure, a value

129

of Q in SHT(r,c) means that Q points in the xy-plane lie on the line specified by

θ(c) and ρ(r). Peak values in the SHT represent potential lines in the input image.

More information about the SHT is referred to JAHNE [99].

Hough transform peaks

θ

-80 -60 -40 -20 0 20 40 60 80

ρ

-1500

-1000

-500

0

500

1000

1500

Figure C.5: Hough transform peaks

In Fig. C.5, the white square shows the identified three peaks in SHT, which

represent three longest lines dectected as shown in Fig. C.6.

Figure C.6: Dectected top three longest line

The water level information is carried by the bottom point of these dectected

lines. As shown in Fig. C.7, the water level now could be extracted from the distance

in pixels between the detected bottom point and the base line. The distance in pixels

can finnaly be converted to meter based on camera calibration.

130

Figure C.7: Calculate the detected water level

131

Appendix D

Uncertainty Analysis of BIV

D.1 Uncertainty analysis of BIV

The uncertainty analysis for a PIV system is referred to ITTC Recommended Proce-

dures and Guidelines (PARK et al. [20]). Since BIV is a direct adoption of the PIV

technique, the uncertainty analysis for a BIV system could be similarly evaluated.

Following the ISO Evaluation of measurement data Guide to the expression

of uncertainty in measurement (BIPM et al. [101]), if an experimental result is

given by the data reduction equation f = f(x1, x2, · · · , xN), the combined standard

uncertainty of the estimate f is denoted by uc(f) and given as,

u2f =

N∑i=1

(∂f

∂xi)2u(xi)

2 (D.1)

where xi is the independent error source of f and u(xi) is the uncertainty of xi.

The principle of the BIV measurement on flow velocity u could be described as,

u = α∆X

∆t+ δu (D.2)

where, α is a magnification factor between physical coordinates (in mm) and image

coordinates (in pixel) and α can be identified through the calibration procedure;

∆X is the displacement of particle in images (in pixel); ∆t is time interval and δu

represents the uncertainty factors of flow visualization. From Eq. (D.2), to evaluate

the uncertainty on u, the uncertainty on the α, ∆X, ∆t, and δu should be evaluated

first.

The calibration was done by using a calibration board. The distance of the

reference point lr and its distance on the image plane Lr were used to determine the

magnification factor, α, as

132

α =lrLrcosθ ≈ lr

Lr(1− θ2

2) (D.3)

where θ is a small angle between the object plane and the calibration board.

Following PARK et al. [20], the error sources and its uncertainty analysis on

θ are summarized in Table D.1. The distance of the reference points (Lr) were

measured on the image plane. If the position of the reference points were detected

from a single point of image, the uncertainty will be 0.5 pixel. Since the distance

is determined by two points, the total amount of uncertainty will be 0.5√

2 = 0.7

pixel. The uncertainties of the physical length of the reference points affect the

accuracy of the magnification factor. Using a well-controlled calibration board, the

uncertainty on lr can be assumed as 20 µm. The image could be distorted by the

aberration of lenses. The distortion of the image affects the error of magnification

factor. The distortion of the image could assumed be as 0.5% of Lr. The uncertainty

band caused by the accuracy of CCD is usually small, and it can be neglected. The

position of the calibration board and the object plane could be different and this

difference is assumed less than 0.5 mm. Ideally, the calibration board should be

parallel to the object plane for visualization. The error due to non-parallel could

assumed to be 2 degree.

Table D.1: Evaluate the uncertainty on α

Error Sources u(xi)∂f∂xi

Lr in pixel 0.5√

2 = 0.7 pixel ∂α∂Lr

= − lrL2r

lr in mm about 20 µm ∂α∂lr

= 1Lr

distortion by lens ≤ 0.5%Lr∂α∂Lr

= − lrL2r

distortion by CCD neglected ∂α∂Lr

Board position lt 0.5 mm ∂α∂lt

= − lrLrlt

Parallel board θ 2◦ = 0.035 rad ∂α∂θ

= − lrLrθ

combined uncertainty uc(α)

The measurement position and time are defined by Eq. D.4 and D.5. Eq. D.4, X0

indicates the location of origin on the image plane, and Xs and Xe show the starting

and ending position of correlation area. The physical location can be obtained by

transferring it with magnification factor α.

x = α(Xs +Xe

2−X0) (D.4)

t =ts + te

2(D.5)

133

The error sources and the uncertainty analysis on image displacement x are

summarized in Table D.2. The spatial and temporal fluctuation of light power could

affect on the detection of particle image position directly. This error is assumed

to be 1/10 of particle diameter dp. The amount of error from camera distortion

could be estimated as 0.0056 pixels. The normal or perpendicular view angle to

the illumination plane could affect the uncertainty of the displacement. The angle

could be estimated as 2 degree. In the pixel unit analysis, mis-matching of pair

particle images could happen and its uncertainty could be estimated as 0.2 pixel.

The uncertainties of sub-pixel analysis depend on many factors such as the diameter

of tracer particle, noise level of the image, and particle concentration. Here, it is

estimated as 0.03 pixel.

Table D.2: Evaluate the uncertainty on image displacement ∆X


Light power fluctuation 110

√2dp

∂X∂x

= 1α

Image distortion by camera 0.0056 pix ∂x∂X

= αNormal view angle 2◦ = 0.035 rad ∂α

∂θ= − lr

Lrθ

Mis-matching error 0.2 pix 1.0Sub-pixel analysis 0.03 pix 1.0

combined uncertainty uc(∆X)

The error sources and the uncertainty analysis on δu are summarized in Table

D.3. The error source of particle trajectory is related to the velocity lag. The error

from velocity lag is assumed to be 0.01% of the total velocity. The error sources of

three-dimensional effects come from the perspective of out-of-plane velocity compo-

nent and assumed as 1.0% of the flow velocity.

Table D.3: Evaluate the uncertainty on δu


Particle trajectory 0.01%u 1.03-D effects 1%utanβ 1.0

combined uncertainty uc(δu)

The error sources and the uncertainty analysis on ∆t are summarized in Table

D.4. The fluctuation of pulse time could be an error source of measurement from

the time series analysis. The uncertainty for it could be 5 ns.

From the calculated combined uncertainty in Table D.1 to Table D.4 and with

Eq. D.2, the uncertainty on u are summarized in Table D.5.

134

Table D.4: Evaluate the uncertainty on ∆t


pulse timing accuracy 5 ns 1.0

combined uncertainty uc(∆t)

Table D.5: Evaluate the uncertainty on u


α uc(α) ∆X∆t

∆X uc(∆X) α∆t

∆t uc(∆t) −α∆X2∆t2

δu uc(δu) 1.0

combined uncertainty uc(u)

D.1.1 Optimize magnification factor

To use the BIV technique, one important consideration is to match the size of a

typical particle image diameter dτ with the resolution of recording medium. The

resolution elements of a video graphic device are the pixel elements and the resolution

is characterized by their spacing. These pixel elements can be considered as discrete

sampling points in the image. If particle image diameter is too small, the image

is under-sampled according to Nyquists criterion and information might be lost.

As demonstrated in RAFFEL et al. [16], the so-called ”peak locking” effects is

introduced when the particle image diameter is too small, which means the bias

error of the system is increased due to under-sampling. On the other hand, if the

image is too large, the system wastes the information capacity by sampling more

often than needed to specify the image.

ADRIAN [79] gived one formulation to optimize the magnification factor by es-

timating the depth of field (DOF). Noticing the formulation for calculating DOF in

ADRIAN [79] is different as the formulation used in RYU et al. [15]. To avoid the

mis-understanding in DOF, here we introduce a new way to optimize the magnifi-

cation factor more directly without calculating the DOF.

According to RAFFEL et al. [16], if lens aberrations can be neglected the particle

image diameter can be estimated as,

dτ =√

(Mdp)2 + d2diff (D.6)

where dp is the particle diameter and ddiff is diffraction limited imaging diameter

135

which could be estimated as

ddiff = 2.44(1 +M)f#λ (D.7)

where f# is the so-called f-number, defined as the ratio of the focal length of a

camera lens to the diameter of the aperture and λ is the wavelength of the light

source.

Assuming dτ can be approximately by,

dτ = crdr (D.8)

where cr donates the number of pixels per particle diameter and dr donates the size

of one pixel. According to ADRIAN [79], the bias error of discrete sampling may

be considered negligible if cr is greater than one to three pixels per diameter, for

reference we shall take cr = 3.

By combining Eq. (D.6), Eq. (D.7) and Eq. (D.8), it follows,

(Mdp)2 + (2.44f#λ)2(1 +M)2 = (crdr)

2 (D.9)

Solving Eq. (D.9) and ignoring the negative solution yields the optimized magnifi-

cation factor,

Mopt =−(2.44f#λ)2 +

√((2.44f#λ)2 + d2

p)(crdr)2 − (2.44f#λ)2d2

p

(2.44f#λ)2 + d2p

(D.10)

It could be seen that f# and dr are constants related to camera properties and λ is

referred to light source. Once the camera and the light source are selected for the

system, the mean particle diameter would be the main consideration for optimizing

the magnification factor.

136

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